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<p><strong>Description:</strong> Gives an overview of probability, including basic definitions, the Monty Hall problem, and strange dice games.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="390">The</span> <span m="520">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5040">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7920">quality</span> <span m="8450">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="12000">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13120">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14790">courses,</span> <span m="15900">visit</span> <span m="16120">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="23170">PROFESSOR: Let's</span> <span m="23380">get</span> <span m="23530">started.</span> <span m="25090">Drop</span> <span m="25440">date</span> <span m="25610">is</span> <span m="25730">coming</span> <span m="26060">up.</span> <span m="26540">So</span> <span m="26840">that</span> <span m="26920">you</span> <span m="27040">didn't</span> <span m="27310">do</span> <span m="27420">very</span> <span m="27660">well</span> <span m="27940">on the</span> <span m="28030">midterm,</span> <span m="28660">and</span> <span m="28810">you</span> <span m="28910">haven't</span> <span m="29210">been</span> <span m="29320">doing</span> <span m="29520">your</span> <span m="29640">homework,</span> <span m="30070">and</span> <span m="30380">you haven't</span> <span m="30630">been</span> <span m="30810">going</span> <span m="30990">to</span> <span m="31060">recitation,</span> <span m="32070">I</span> <span m="32170">would</span> <span m="32310">strongly</span> <span m="32729">consider</span> <span m="33930">drop</span> <span m="34210">date.</span> <span m="35580">And</span> <span m="35700">you</span> <span m="35790">can</span> <span m="35910">see</span> <span m="36180">your</span> <span m="36330">TAs</span> <span m="37130">to</span> <span m="37220">get</span> <span m="37380">some</span> <span m="37580">idea</span> <span m="37820">of</span> <span m="37880">what</span> <span m="38020">your</span> <span m="38130">grades</span> <span m="38560">are,</span> <span m="38870">and</span> <span m="39700">what</span> <span m="39920">you're</span> <span m="40040">likely</span> <span m="40430">to</span> <span m="40500">do</span> <span m="40690">if</span> <span m="40770">you</span> <span m="40900">keep</span> <span m="41140">on</span> <span m="42360">giving</span> <span m="42610">the</span> <span m="42710">same</span> <span m="42940">performance</span> <span m="43410">you've</span> <span m="43530">had</span> <span m="43800">so</span> <span m="43980">far.</span> </p>
<p><span m="46400">Now,</span> <span m="46590">today,</span> <span m="47160">we're</span> <span m="47350">going</span> <span m="47450">to</span> <span m="47700">switch</span> <span m="48030">gears</span> <span m="48310">again</span> <span m="49160">and</span> <span m="49500">cover</span> <span m="49740">the</span> <span m="49850">last</span> <span m="50330">topic</span> <span m="50670">for</span> <span m="50750">the</span> <span m="50850">course.</span> <span m="52070">For</span> <span m="52090">the</span> <span m="52180">remainder</span> <span m="52550">of</span> <span m="52650">the</span> <span m="52670">course,</span> <span m="53050">we're</span> <span m="53160">going</span> <span m="53240">to</span> <span m="53310">talk</span> <span m="53560">about</span> <span m="53950">probability.</span> <span m="55800">It's</span> <span m="56250">one</span> <span m="56470">of</span> <span m="56510">the</span> <span m="56580">most</span> <span m="56880">important</span> <span m="57460">subjects</span> <span m="58340">in</span> <span m="58520">all</span> <span m="58880">of</span> <span m="58990">math</span> <span m="59440">or</span> <span m="59540">computer</span> <span m="59940">science.</span> <span m="60540">In</span> <span m="60640">fact,</span> <span m="60860">I</span> <span m="60910">was</span> <span m="61600">at</span> <span m="61790">a</span> <span m="61840">frat</span> <span m="62160">last</span> <span m="62440">night</span> <span m="62620">for</span> <span m="62720">dinner,</span> <span m="63020">and</span> <span m="63100">they</span> <span m="63220">asked</span> <span m="63540">me</span> <span m="63670">what</span> <span m="64550">my</span> <span m="64670">favorite</span> <span m="65090">algorithm</span> <span m="65310">was.</span> </p>
<p><span m="65550">And</span> <span m="65960">I</span> <span m="66490">couldn't</span> <span m="66660">really</span> <span m="66920">think</span> <span m="67240">about</span> <span m="67510">my</span> <span m="67620">favorite</span> <span m="67970">algorithm</span> <span m="68340">was,</span> <span m="68730">but</span> <span m="68840">in</span> <span m="68920">terms</span> <span m="69220">of</span> <span m="69300">one</span> <span m="69430">of</span> <span m="69470">the</span> <span m="69530">most</span> <span m="69740">important</span> <span m="70210">subjects</span> <span m="71130">for</span> <span m="71250">you</span> <span m="71370">to</span> <span m="71470">know</span> <span m="71720">about,</span> <span m="72340">after</span> <span m="72730">induction,</span> <span m="73690">it's probably</span> <span m="73850">probability.</span> <span m="75730">It's</span> <span m="75900">really</span> <span m="76270">important</span> <span m="76710">across</span> <span m="77110">math</span> <span m="77380">and</span> <span m="77450">computer</span> <span m="77800">science.</span> <span m="78850">Most</span> <span m="79250">of</span> <span m="79320">the</span> <span m="79460">upper</span> <span m="79660">level</span> <span m="79960">courses</span> <span m="80360">you</span> <span m="80510">take</span> <span m="80820">in</span> <span m="80870">computer</span> <span m="81300">science</span> <span m="81830">are</span> <span m="81980">going</span> <span m="82100">to</span> <span m="82140">use</span> <span m="82440">probability.</span> <span m="83020">They're</span> <span m="83160">going</span> <span m="83260">to</span> <span m="83320">expect</span> <span m="83720">you</span> <span m="83810">to</span> <span m="83910">know</span> <span m="84080">the</span> <span m="84180">basics.</span> </p>
<p><span m="85660">For</span> <span m="85760">example,</span> <span m="86320">when</span> <span m="86420">you're</span> <span m="86450">designing</span> <span m="86930">algorithms</span> <span m="87560">in</span> <span m="88100">006</span> <span m="89110">or</span> <span m="89310">6046,</span> <span m="91510">you're</span> <span m="91620">going</span> <span m="91720">to</span> <span m="91770">be</span> <span m="91970">designing</span> <span m="92390">in</span> <span m="92460">some</span> <span m="92730">cases</span> <span m="93360">probabilistic</span> <span m="94270">algorithms--</span> <span m="95390">algorithms</span> <span m="96070">that</span> <span m="96250">flip</span> <span m="96540">coins</span> <span m="96990">or</span> <span m="97070">generate</span> <span m="97460">random</span> <span m="97830">numbers</span> <span m="98620">to</span> <span m="98710">get</span> <span m="98920">the</span> <span m="99010">result.</span> <span m="99910">And</span> <span m="100060">by</span> <span m="100170">using</span> <span m="100500">randomness,</span> <span m="101520">they</span> <span m="101660">run</span> <span m="102010">faster.</span> <span m="103410">If</span> <span m="103530">you're</span> <span m="103650">doing</span> <span m="103980">software</span> <span m="104510">engineering</span> <span m="105050">later,</span> <span m="105830">you're</span> <span m="105970">going</span> <span m="106090">to</span> <span m="106130">use</span> <span m="106550">probability</span> <span m="107170">when</span> <span m="107300">you</span> <span m="107390">analyze</span> <span m="108010">system</span> <span m="108370">performance</span> <span m="109040">or</span> <span m="109300">build</span> <span m="109700">hash</span> <span m="110000">tables.</span> <span m="111950">Probability</span> <span m="112560">is</span> <span m="112650">used in</span> <span m="113110">information</span> <span m="113810">theory,</span> <span m="114190">in</span> <span m="114610">coding</span> <span m="115270">theory,</span> <span m="115990">in</span> <span m="116370">cryptography.</span> </p>
<p><span m="117590">In</span> <span m="117710">fact,</span> <span m="118370">our</span> <span m="118510">nation's</span> <span m="119180">codes,</span> <span m="120300">their</span> <span m="120500">security</span> <span m="121150">depends</span> <span m="121740">on</span> <span m="122100">probabilistic</span> <span m="123070">analysis.</span> <span m="124150">Because</span> <span m="124480">the</span> <span m="124550">bad</span> <span m="124850">guys</span> <span m="125560">are</span> <span m="125650">using</span> <span m="125960">probabilistic</span> <span m="126580">algorithms</span> <span m="127310">to</span> <span m="127610">break</span> <span m="128060">the</span> <span m="128139">crypto-system.</span> <span m="129199">And</span> <span m="129330">we'll</span> <span m="129430">give</span> <span m="129610">an</span> <span m="129680">example</span> <span m="130139">of</span> <span m="130229">that</span> <span m="130630">in</span> <span m="130750">another</span> <span m="131000">week</span> <span m="131220">or</span> <span m="131310">so.</span> </p>
<p><span m="132920">Probability</span> <span m="133480">is</span> <span m="133610">obviously</span> <span m="134340">important</span> <span m="134820">if</span> <span m="134920">you're</span> <span m="135030">doing</span> <span m="135310">game</span> <span m="135610">theory</span> <span m="136060">or</span> <span m="136280">if</span> <span m="136340">you</span> <span m="136440">like</span> <span m="136650">to</span> <span m="136740">gamble,</span> <span m="138240">really</span> <span m="138560">critical</span> <span m="138950">in</span> <span m="139050">gambling.</span> <span m="139480">In</span> <span m="139550">fact,</span> <span m="140290">we're</span> <span m="140440">going</span> <span m="140520">to</span> <span m="140570">analyze</span> <span m="140990">a</span> <span m="141060">lot</span> <span m="141390">of</span> <span m="141460">different</span> <span m="141770">gambling</span> <span m="142160">games</span> <span m="142460">over</span> <span m="142570">the</span> <span m="142640">next</span> <span m="142870">few</span> <span m="143030">weeks.</span> <span m="143350">And</span> <span m="143500">for</span> <span m="143610">fun,</span> <span m="144430">we'll</span> <span m="144590">play</span> <span m="144870">some</span> <span m="145140">in</span> <span m="145230">class.</span> <span m="145710">In</span> <span m="145800">fact,</span> <span m="146170">we'll</span> <span m="146300">play</span> <span m="146520">one</span> <span m="146950">later</span> <span m="147220">today.</span> <span m="148170">And</span> <span m="148400">it's</span> <span m="148540">amazing</span> <span m="148980">how</span> <span m="149200">often</span> <span m="150710">the</span> <span m="150830">intuition</span> <span m="151300">is</span> <span m="151400">wrong.</span> <span m="152010">You</span> <span m="152170">think</span> <span m="152400">the</span> <span m="152530">answer</span> <span m="152780">goes</span> <span m="152960">one</span> <span m="153180">way,</span> <span m="153440">and</span> <span m="153540">then</span> <span m="153710">it's</span> <span m="154020">actually</span> <span m="154740">the</span> <span m="154910">other.</span> </p>
<p><span m="156000">Probability</span> <span m="156630">is</span> <span m="156750">important</span> <span m="157220">in</span> <span m="157300">the</span> <span m="157390">study</span> <span m="157690">of</span> <span m="157780">fault</span> <span m="158050">tolerance.</span> <span m="159160">For</span> <span m="159260">example,</span> <span m="160270">how</span> <span m="160340">do</span> <span m="160400">you</span> <span m="160510">figure</span> <span m="160900">out</span> <span m="161090">the</span> <span m="161170">probability</span> <span m="161820">there's</span> <span m="162020">going</span> <span m="162130">to</span> <span m="162170">be</span> <span m="162300">some</span> <span m="162610">critical</span> <span m="163080">failure</span> <span m="163600">that</span> <span m="163730">causes</span> <span m="164150">the</span> <span m="164250">space</span> <span m="164610">shuttle</span> <span m="164970">to</span> <span m="165080">crash?</span> <span m="166120">Or</span> <span m="166500">a</span> <span m="166540">critical</span> <span m="166910">failure</span> <span m="167270">that</span> <span m="167410">causes</span> <span m="168120">a</span> <span m="168240">nuclear</span> <span m="168640">plant</span> <span m="168990">to</span> <span m="169080">meltdown?</span> <span m="170350">That's</span> <span m="170610">done</span> <span m="170890">through</span> <span m="171030">probabilistic</span> <span m="171630">analysis.</span> <span m="172250">And</span> <span m="172370">we'll</span> <span m="172680">do</span> <span m="172820">some</span> <span m="173000">examples</span> <span m="173500">of</span> <span m="173570">that.</span> </p>
<p><span m="174780">Probability</span> <span m="175460">comes</span> <span m="175770">up</span> <span m="176120">in</span> <span m="176260">everyday</span> <span m="176700">life.</span> <span m="177370">We</span> <span m="177490">just</span> <span m="177690">had</span> <span m="177850">an</span> <span m="177930">election.</span> <span m="179160">And</span> <span m="179500">running</span> <span m="179820">up</span> <span m="179940">to the</span> <span m="180070">election,</span> <span m="180490">you</span> <span m="180570">get</span> <span m="180680">all</span> <span m="180880">these</span> <span m="181100">polls</span> <span m="182250">that</span> <span m="182430">say</span> <span m="182890">that's</span> <span m="183130">so-and-so</span> <span m="183780">is</span> <span m="183890">ahead</span> <span m="184200">of</span> <span m="184340">so-and-so</span> <span m="185520">by</span> <span m="185690">three</span> <span m="186010">points</span> <span m="186520">with</span> <span m="186660">a</span> <span m="187660">margin</span> <span m="187990">of error</span> <span m="188060">of</span> <span m="188410">five</span> <span m="188760">points.</span> <span m="189780">So</span> <span m="190090">it's</span> <span m="190210">a</span> <span m="190270">statistical</span> <span m="190820">dead</span> <span m="191020">heat,</span> <span m="191400">is</span> <span m="191510">what</span> <span m="191670">they</span> <span m="191800">say.</span> <span m="192650">Well,</span> <span m="192800">what</span> <span m="192900">the</span> <span m="192980">heck</span> <span m="193180">does</span> <span m="193310">that</span> <span m="193570">mean?</span> <span m="194930">In</span> <span m="194990">fact,</span> <span m="195340">usually</span> <span m="195590">they've</span> <span m="195690">got</span> <span m="195860">it</span> <span m="195970">screwed</span> <span m="196290">up.</span> <span m="196990">And</span> <span m="197170">we're</span> <span m="197280">going</span> <span m="197370">to</span> <span m="197440">spend</span> <span m="197730">the</span> <span m="197830">recitation</span> <span m="198550">where</span> <span m="198660">you're</span> <span m="198800">going</span> <span m="198930">to</span> <span m="199080">do</span> <span m="199390">a</span> <span m="199470">margin</span> <span m="199950">of</span> <span m="200050">error</span> <span m="200880">analysis</span> <span m="201710">for</span> <span m="201940">a</span> <span m="201970">poll,</span> <span m="202880">and</span> <span m="203030">see</span> <span m="203190">how</span> <span m="203350">many</span> <span m="203560">people</span> <span m="203870">do you</span> <span m="204010">need</span> <span m="204260">to</span> <span m="204340">poll</span> <span m="205240">to</span> <span m="205350">get</span> <span m="205500">a</span> <span m="205560">pretty</span> <span m="205910">accurate</span> <span m="206630">result?</span> </p>
<p><span m="208490">Comes</span> <span m="208820">up</span> <span m="208980">in</span> <span m="209200">the</span> <span m="209320">medical</span> <span m="209700">studies.</span> <span m="211390">Does</span> <span m="211640">cholesterol</span> <span m="212800">cause</span> <span m="213130">heart</span> <span m="213340">disease?</span> <span m="214400">Does</span> <span m="214560">smoking</span> <span m="215150">cause</span> <span m="215610">lung</span> <span m="215830">cancer?</span> <span m="217090">How</span> <span m="217410">effective</span> <span m="217880">is</span> <span m="217990">a</span> <span m="218060">certain</span> <span m="218370">drug?</span> <span m="219410">Say you</span> <span m="219740">get</span> <span m="220000">tested</span> <span m="220470">for</span> <span m="220640">a</span> <span m="220740">disease</span> <span m="221240">and</span> <span m="221330">it</span> <span m="221400">comes</span> <span m="221670">back</span> <span m="221940">positive.</span> <span m="223780">Well,</span> <span m="224120">there's</span> <span m="224360">false</span> <span m="224670">positives</span> <span m="225130">and</span> <span m="225250">false</span> <span m="225570">negatives.</span> <span m="225950">What</span> <span m="226110">are</span> <span m="226130">the</span> <span m="226260">chances</span> <span m="226700">you</span> <span m="226860">really</span> <span m="227110">have</span> <span m="227810">the</span> <span m="227910">disease?</span> <span m="228970">So</span> <span m="229110">the</span> <span m="229200">list</span> <span m="229560">goes</span> <span m="229800">on</span> <span m="230010">and</span> <span m="230120">on.</span> <span m="231960">In</span> <span m="232090">fact,</span> <span m="232320">we're</span> <span m="232400">going</span> <span m="232490">to</span> <span m="232880">see</span> <span m="233090">all</span> <span m="233290">these</span> <span m="233460">examples.</span> <span m="233900">We've</span> <span m="233980">got</span> <span m="234170">eight</span> <span m="234380">lectures</span> <span m="234850">and eight</span> <span m="235120">recitations.</span> <span m="235910">And</span> <span m="236020">there's</span> <span m="236140">just</span> <span m="236360">no</span> <span m="236670">end</span> <span m="236960">of</span> <span m="237160">interesting</span> <span m="237480">examples</span> <span m="238020">where</span> <span m="238100">probability</span> <span m="238650">comes</span> <span m="238900">into</span> <span m="239070">play.</span> </p>
<p><span m="240770">The</span> <span m="240920">bad</span> <span m="241300">news</span> <span m="241590">is,</span> <span m="242250">probability</span> <span m="243680">is</span> <span m="244020">probably</span> <span m="244920">the</span> <span m="245030">most</span> <span m="245450">misunderstood</span> <span m="246910">subject</span> <span m="248070">in</span> <span m="248210">all</span> <span m="248460">of</span> <span m="248560">mathematics</span> <span m="249400">and</span> <span m="249540">the</span> <span m="249610">sciences.</span> <span m="251370">In</span> <span m="251480">fact,</span> <span m="252780">Mark</span> <span m="253020">Twain</span> <span m="253520">once</span> <span m="253870">said,</span> <span m="255370">there's</span> <span m="255750">three</span> <span m="256089">kinds</span> <span m="256500">of</span> <span m="256630">lies--</span> <span m="257890">lies,</span> <span m="259510">damned</span> <span m="259990">lies,</span> <span m="261560">and</span> <span m="261680">statistics,</span> <span m="263300">meaning</span> <span m="263610">probability.</span> <span m="265210">Because</span> <span m="265520">it's</span> <span m="265820">so</span> <span m="266180">easy</span> <span m="267650">to</span> <span m="267800">lie</span> <span m="269000">with</span> <span m="269200">statistics.</span> <span m="269890">And</span> <span m="270000">we're</span> <span m="270110">going</span> <span m="270190">to</span> <span m="270250">see</span> <span m="270420">lots</span> <span m="270750">and</span> <span m="271090">lots</span> <span m="271330">of</span> <span m="271410">examples</span> <span m="271900">of</span> <span m="271960">that.</span> </p>
<p><span m="273750">In</span> <span m="273880">fact,</span> <span m="274780">newspaper</span> <span m="275440">articles,</span> <span m="276390">you</span> <span m="276510">know,</span> <span m="276790">have</span> <span m="276990">all</span> <span m="277300">sorts</span> <span m="277600">of</span> <span m="277720">things</span> <span m="278100">that</span> <span m="278210">are</span> <span m="278270">just</span> <span m="278970">horribly</span> <span m="279400">fallacious</span> <span m="280000">based</span> <span m="280410">on</span> <span m="280520">statistics.</span> <span m="281050">And</span> <span m="281130">we'll</span> <span m="281270">see</span> <span m="281470">examples.</span> <span m="281990">In</span> <span m="282090">fact,</span> <span m="282770">we're</span> <span m="283130">going</span> <span m="283240">to</span> <span m="283280">even</span> <span m="283620">debunk</span> <span m="284150">a</span> <span m="284250">very</span> <span m="284650">famous</span> <span m="286180">article--</span> <span m="286690">published</span> <span m="287130">research</span> <span m="287530">paper--</span> <span m="287830">by</span> <span m="287980">some</span> <span m="288160">famous</span> <span m="288550">computer</span> <span m="288920">scientists</span> <span m="289510">at</span> <span m="289590">Berkeley.</span> <span m="290720">Where</span> <span m="290940">they</span> <span m="291080">got</span> <span m="291310">to</span> <span m="291410">a</span> <span m="291450">conclusion</span> <span m="292070">using</span> <span m="292320">a</span> <span m="292350">very</span> <span m="292850">standard</span> <span m="293310">method</span> <span m="293610">you</span> <span m="293690">see</span> <span m="293870">all</span> <span m="293960">the</span> <span m="294060">time--</span> <span m="294700">completely</span> <span m="295320">fallacious.</span> <span m="296650">And</span> <span m="296800">we'll</span> <span m="296930">see</span> <span m="297110">that.</span> </p>
<p><span m="298120">So</span> <span m="298420">you</span> <span m="298530">can</span> <span m="298670">think</span> <span m="298830">of</span> <span m="298880">that</span> <span m="299070">as,</span> <span m="299570">I'm</span> <span m="299720">going</span> <span m="299810">to</span> <span m="299900">teach</span> <span m="300160">you</span> <span m="300280">how</span> <span m="300410">to</span> <span m="300490">lie</span> <span m="301810">with</span> <span m="301990">statistics.</span> <span m="303080">Even</span> <span m="303310">better,</span> <span m="304070">how</span> <span m="304270">not</span> <span m="304720">to</span> <span m="304810">be</span> <span m="305080">taken</span> <span m="305420">in</span> <span m="305610">by</span> <span m="306080">incorrect</span> <span m="306640">arguments</span> <span m="307580">using</span> <span m="308380">probability</span> <span m="309010">theory.</span> <span m="310860">Now,</span> <span m="311760">at</span> <span m="311900">MIT,</span> <span m="313240">it's</span> <span m="313470">probably</span> <span m="314000">the</span> <span m="314110">subject</span> <span m="315390">that</span> <span m="316250">most</span> <span m="316630">often</span> <span m="316990">trips</span> <span m="317290">up</span> <span m="317410">students</span> <span m="317910">in</span> <span m="318030">their</span> <span m="318190">PhD</span> <span m="318960">qualifying</span> <span m="319490">exams.</span> <span m="320760">You know,</span> <span m="321110">there's</span> <span m="321400">no</span> <span m="321670">better</span> <span m="322040">way</span> <span m="322740">to</span> <span m="322960">strike</span> <span m="323450">terror</span> <span m="324210">in</span> <span m="324340">the</span> <span m="324420">heart</span> <span m="324640">of</span> <span m="324720">a</span> <span m="324790">student</span> <span m="325720">then</span> <span m="325950">to</span> <span m="326580">be</span> <span m="326690">sitting</span> <span m="326980">there</span> <span m="327140">and</span> <span m="327220">ask</span> <span m="327430">them</span> <span m="327550">a</span> <span m="327600">question</span> <span m="327970">while</span> <span m="328270">they're</span> <span m="328650">at</span> <span m="328730">the</span> <span m="328810">board,</span> <span m="329830">what's</span> <span m="330070">the</span> <span m="330170">probability</span> <span m="330860">that</span> <span m="331220">foo</span> <span m="331510">happens?</span> <span m="332480">It's</span> <span m="332660">like</span> <span m="332970">panic</span> <span m="333510">sets</span> <span m="333790">in</span> <span m="334310">when</span> <span m="334480">they've</span> <span m="334610">got</span> <span m="334760">to</span> <span m="334820">do</span> <span m="334920">a</span> <span m="335000">probabilistic</span> <span m="335480">analysis.</span> </p>
<p><span m="336060">And</span> <span m="336140">now,</span> <span m="336630">MIT</span> <span m="337090">PhD</span> <span m="337610">students</span> <span m="338320">are</span> <span m="338480">good.</span> <span m="339330">You know,</span> <span m="339660">they</span> <span m="339810">can</span> <span m="339980">crank</span> <span m="340360">out</span> <span m="340720">the</span> <span m="340780">most</span> <span m="341030">horrendous</span> <span m="341500">calculation,</span> <span m="342260">no</span> <span m="342420">problem,</span> <span m="343520">come</span> <span m="343670">up</span> <span m="343780">with</span> <span m="343880">really</span> <span m="344150">clever</span> <span m="344510">ideas</span> <span m="344950">to</span> <span m="345050">solve</span> <span m="345320">problems.</span> <span m="345880">But</span> <span m="345930">you</span> <span m="346040">get</span> <span m="346220">into</span> <span m="346370">probability,</span> <span m="347850">it</span> <span m="347980">gets</span> <span m="348230">into</span> <span m="348380">trouble.</span> <span m="349170">In</span> <span m="349270">fact,</span> <span m="349590">even</span> <span m="349840">the</span> <span m="349910">other</span> <span m="350070">faculty</span> <span m="350550">sitting</span> <span m="350890">there</span> <span m="351070">start</span> <span m="351330">squirming</span> <span m="351870">around</span> <span m="353010">when</span> <span m="353180">this</span> <span m="353360">question's</span> <span m="353820">being</span> <span m="354090">posed.</span> <span m="355050">You</span> <span m="355160">know,</span> <span m="355480">and</span> <span m="357020">probability</span> <span m="357540">questions</span> <span m="357990">are--</span> <span m="358100">and we'll</span> <span m="358400">see</span> <span m="358560">some</span> <span m="358680">examples</span> <span m="359120">today--</span> <span m="359430">are</span> <span m="359560">always</span> <span m="359810">the</span> <span m="359900">best</span> <span m="360260">at</span> <span m="360460">faculty</span> <span m="360880">parties,</span> <span m="361740">just</span> <span m="361980">can</span> <span m="362140">ruin</span> <span m="362460">people's</span> <span m="362790">whole</span> <span m="363060">night.</span> <span m="363450">You</span> <span m="363540">know,</span> <span m="363710">what's</span> <span m="363890">the</span> <span m="363980">probability</span> <span m="364430">this</span> <span m="364610">happens?</span> <span m="365000">And it's</span> <span m="365200">some</span> <span m="365470">weird</span> <span m="366250">answer.</span> </p>
<p><span m="369180">Now,</span> <span m="369810">the</span> <span m="369910">strange</span> <span m="370380">thing</span> <span m="370560">about</span> <span m="370870">this--</span> <span m="371040">well,</span> <span m="371130">first</span> <span m="371390">strange thing</span> <span m="371740">is</span> <span m="371870">that</span> <span m="372090">our</span> <span m="372280">human</span> <span m="372600">intuition</span> <span m="373150">is</span> <span m="373250">terrible.</span> <span m="373820">And</span> <span m="373960">you</span> <span m="374080">will</span> <span m="374220">see</span> <span m="374440">that.</span> <span m="375380">Your</span> <span m="375670">intuition</span> <span m="376220">will</span> <span m="376450">be</span> <span m="376590">wrong,</span> <span m="377510">just</span> <span m="377730">the</span> <span m="377810">way</span> <span m="377960">our</span> <span m="378090">brains</span> <span m="378610">are</span> <span m="378800">created</span> <span m="379870">genetically.</span> <span m="381840">But</span> <span m="381880">the</span> <span m="381960">good</span> <span m="382180">news</span> <span m="382420">is</span> <span m="382810">that</span> <span m="382920">probability</span> <span m="383560">can</span> <span m="384110">be</span> <span m="384220">very</span> <span m="384620">simple.</span> <span m="385880">If</span> <span m="385950">you</span> <span m="386090">force</span> <span m="386630">yourself</span> <span m="387650">to</span> <span m="387880">just</span> <span m="388210">go</span> <span m="388350">to</span> <span m="388410">the</span> <span m="388510">basic</span> <span m="388930">principles--</span> <span m="389710">which</span> <span m="389880">we're</span> <span m="390060">going</span> <span m="390150">to</span> <span m="390220">cover--</span> <span m="391140">and</span> <span m="391340">go through</span> <span m="391610">the</span> <span m="391720">basic</span> <span m="392480">template</span> <span m="392920">of</span> <span m="392980">how</span> <span m="393190">to</span> <span m="393290">solve</span> <span m="393540">a</span> <span m="393610">problem--</span> <span m="394230">which</span> <span m="394410">we're</span> <span m="394540">going</span> <span m="394630">to</span> <span m="394690">cover--</span> <span m="395630">it</span> <span m="395760">is</span> <span m="395910">easy,</span> <span m="397030">and</span> <span m="397210">you</span> <span m="397300">can't</span> <span m="397580">go</span> <span m="397740">wrong.</span> </p>
<p><span m="399370">The</span> <span m="400120">key</span> <span m="400950">is</span> <span m="401110">forcing</span> <span m="401550">yourself</span> <span m="401910">to</span> <span m="402000">just</span> <span m="402280">actually</span> <span m="403300">write</span> <span m="403560">it</span> <span m="403660">down,</span> <span m="404060">follow</span> <span m="404420">the</span> <span m="404500">basic</span> <span m="404810">principles,</span> <span m="405850">and</span> <span m="406050">get</span> <span m="406210">to</span> <span m="406290">the</span> <span m="406440">answer.</span> <span m="407110">As</span> <span m="407280">opposed</span> <span m="407640">to</span> <span m="407680">saying,</span> <span m="408150">oh,</span> <span m="409540">the</span> <span m="409680">answer's</span> <span m="410020">that</span> <span m="410530">and</span> <span m="410620">here's</span> <span m="410830">why.</span> <span m="412030">Because</span> <span m="412220">it's</span> <span m="412350">amazing</span> <span m="412780">how</span> <span m="412950">often</span> <span m="413220">you'll</span> <span m="413340">be</span> <span m="413460">wrong</span> <span m="413910">if</span> <span m="414010">you</span> <span m="414270">just</span> <span m="414480">try</span> <span m="414620">to</span> <span m="414720">rely</span> <span m="415490">on</span> <span m="415650">your</span> <span m="415800">intuition.</span> </p>
<p><span m="417620">Now,</span> <span m="417760">we're</span> <span m="417910">going</span> <span m="418010">to</span> <span m="418080">start</span> <span m="418580">today</span> <span m="419210">our</span> <span m="419350">study</span> <span m="419610">of</span> <span m="419680">probability</span> <span m="420730">with</span> <span m="422280">a</span> <span m="422370">famous,</span> <span m="423110">simple</span> <span m="423490">game.</span> <span m="424760">And</span> <span m="425090">the</span> <span m="425160">game</span> <span m="425510">is</span> <span m="425660">known</span> <span m="426020">as</span> <span m="426170">the</span> <span m="426360">Monty</span> <span m="426810">Hall</span> <span m="427130">problem.</span> <span m="428270">Now,</span> <span m="428840">do you</span> <span m="429000">all</span> <span m="429140">know who</span> <span m="429400">Monty</span> <span m="429820">Hall</span> <span m="430150">was?</span> <span m="431500">You're</span> <span m="431640">probably</span> <span m="432120">too</span> <span m="432300">young.</span> <span m="432590">Yeah,</span> <span m="432960">OK,</span> <span m="433180">not</span> <span m="433370">many</span> <span m="433540">people</span> <span m="433820">know who</span> <span m="434010">Monte</span> <span m="434370">Hall was.</span> <span m="434850">But back</span> <span m="435100">in</span> <span m="435200">the</span> <span m="435280">1970s,</span> <span m="436820">there</span> <span m="436950">was</span> <span m="437110">a</span> <span m="437290">very</span> <span m="437670">popular</span> <span m="438150">daytime</span> <span m="439150">TV</span> <span m="439530">show--</span> <span m="439790">a</span> <span m="439860">game</span> <span m="440170">show--</span> <span m="441030">called</span> <span m="441500"><i>Let's</i></span> <span m="441860"><i>Make</i></span> <span m="442100"><i>a</i></span> <span m="442170"><i>Deal.</i></span> <span m="444850">And</span> <span m="445530">it was</span> <span m="445620">a</span> <span m="445680">great</span> <span m="445950">show.</span> <span m="446090">It was</span> <span m="446200">the</span> <span m="446280">precursor</span> <span m="446890">of</span> <span m="446980"><i>Wheel</i></span> <span m="447220"><i>of</i></span> <span m="447300"><i>Fortune--</i></span> <span m="447870">if</span> <span m="447970">you're</span> <span m="448610">old enough</span> <span m="448800">to have</span> <span m="449120">seen</span> <span m="449410">that--</span> <span m="449940">and</span> <span m="450090">then</span> <span m="450240"><i>Deal</i></span> <span m="450520"><i>or</i></span> <span m="450580"><i>No</i></span> <span m="450800"><i>Deal</i></span> <span m="451300">after</span> <span m="451620">that.</span> <span m="451900">There's</span> <span m="452050">always</span> <span m="452270">some</span> <span m="452600">game</span> <span m="452860">show</span> <span m="453060">that's</span> <span m="453250">really</span> <span m="453500">popular.</span> </p>
<p><span m="454500">Anyway,</span> <span m="454830">back</span> <span m="455160">in</span> <span m="455240">the</span> <span m="455320">'70s,</span> <span m="455920">it</span> <span m="456030">was</span> <span m="456250"><i>Let's</i></span> <span m="456460"><i>Make</i></span> <span m="456650"><i>a</i></span> <span m="456720"><i>Deal,</i></span> <span m="457660">and</span> <span m="458830">Monty</span> <span m="459180">Hall</span> <span m="459390">was</span> <span m="459510">the</span> <span m="459600">host.</span> <span m="460390">How</span> <span m="460520">many</span> <span m="460710">people</span> <span m="461010">have</span> <span m="461260">ever</span> <span m="461480">heard</span> <span m="461670">of</span> <span m="461740"><i>Let's</i></span> <span m="461910"><i>Make</i></span> <span m="462080">a</span> <span m="462120"><i>Deal?</i></span> <span m="462340">Raise</span> <span m="462510">your</span> <span m="462630">hand</span> <span m="462830">if</span> <span m="462910">you've</span> <span m="462970">ever</span> <span m="463120">heard</span> <span m="463280">of</span> <span m="463320">that.</span> <span m="463910">OK,</span> <span m="464390">so</span> <span m="464550">it</span> <span m="464760">still</span> <span m="464980">has</span> <span m="465150">some fame.</span> <span m="465630">Good.</span> <span m="466460">All</span> <span m="466550">right.</span> <span m="467690">Now,</span> <span m="468960">Monty</span> <span m="469330">Hall,</span> <span m="469670">of course,</span> <span m="469990">was</span> <span m="470170">the</span> <span m="470300">precursor</span> <span m="470970">of</span> <span m="471100">Pat Sajak</span> <span m="472100">of</span> <span m="472210"><i>Wheel</i></span> <span m="472400"><i>of</i></span> <span m="472490"><i>Fortune</i></span> <span m="473350">and</span> <span m="473470">Howie</span> <span m="473690">Mandel,</span> <span m="474740">you know,</span> <span m="474860">from</span> <span m="474945"><i>Deal</i></span> <span m="475030"><i>or</i></span> <span m="475130"><i>No</i></span> <span m="475310"><i>Deal.</i></span> <span m="476410">And</span> <span m="476960">his</span> <span m="477180">assistant</span> <span m="477700">was</span> <span m="478190">Carol</span> <span m="478540">Merrill.</span> <span m="479070">She</span> <span m="479250">was</span> <span m="479420">also</span> <span m="479690">famous</span> <span m="480120">as</span> <span m="480280">the</span> <span m="480990">beautiful</span> <span m="481430">assistant.</span> </p>
<p><span m="482620">Now,</span> <span m="482930">in</span> <span m="483030"><i>Let's</i></span> <span m="483140"><i>Make</i></span> <span m="483340"><i>a</i></span> <span m="483380"><i>Deal,</i></span> <span m="484300">they</span> <span m="484420">TV</span> <span m="484880">audience</span> <span m="485430">would</span> <span m="485580">come</span> <span m="485860">dressed</span> <span m="486320">up</span> <span m="486500">in</span> <span m="486750">wild</span> <span m="487200">costumes.</span> <span m="488890">And</span> <span m="489490">Monty</span> <span m="490280">would</span> <span m="490490">go</span> <span m="491350">into</span> <span m="491610">the</span> <span m="491760">audience</span> <span m="492690">and</span> <span m="492980">pick</span> <span m="493240">out</span> <span m="493390">a</span> <span m="493440">contestant</span> <span m="494080">from</span> <span m="494210">the</span> <span m="494340">audience</span> <span m="494890">based</span> <span m="495240">on</span> <span m="495370">how</span> <span m="495610">wild</span> <span m="495880">their</span> <span m="496150">costume</span> <span m="496690">was--</span> <span m="497460">you</span> <span m="497530">know,</span> <span m="497620">somebody</span> <span m="497950">that</span> <span m="498040">would</span> <span m="498200">look</span> <span m="498350">interesting</span> <span m="498740">for</span> <span m="498860">TV.</span> <span m="499910">And</span> <span m="500170">what</span> <span m="500290">he would</span> <span m="500480">do</span> <span m="500720">is</span> <span m="500820">he'd</span> <span m="500920">buy</span> <span m="501340">a piece</span> <span m="501430">of</span> <span m="501740">the</span> <span m="501890">costume--</span> <span m="502440">maybe</span> <span m="502970">their</span> <span m="503130">hat or</span> <span m="504410">shirt</span> <span m="504810">or</span> <span m="504910">something--</span> <span m="505290">and</span> <span m="505400">give</span> <span m="505510">them</span> <span m="505650">$100</span> <span m="506340">for</span> <span m="506470">a</span> <span m="506540">piece</span> <span m="506790">of</span> <span m="506830">their</span> <span m="506940">costume.</span> <span m="507960">And</span> <span m="508220">then</span> <span m="508390">he</span> <span m="508520">would</span> <span m="508690">say,</span> <span m="509980">hey</span> <span m="510140">look,</span> <span m="510510">you</span> <span m="510620">know,</span> <span m="511370">if</span> <span m="511490">you</span> <span m="511660">give</span> <span m="511820">me</span> <span m="511960">back the</span> <span m="512360">$100</span> <span m="513039">I</span> <span m="513130">just</span> <span m="513380">gave</span> <span m="513640">you,</span> <span m="514340">I'm</span> <span m="514530">going</span> <span m="514650">to</span> <span m="514700">let</span> <span m="514950">you</span> <span m="515120">have</span> <span m="515380">your</span> <span m="515549">choice</span> <span m="516419">of</span> <span m="516960">what's</span> <span m="517230">behind</span> <span m="517659">door</span> <span m="517880">number</span> <span m="518120">one,</span> <span m="518630">door</span> <span m="518830">number</span> <span m="519090">two,</span> <span m="519350">or</span> <span m="519450">door</span> <span m="519669">number</span> <span m="519909">three.</span> <span m="520730">Or</span> <span m="520870">sometimes</span> <span m="521330">it</span> <span m="521390">would</span> <span m="521520">be</span> <span m="521650">box</span> <span m="522059">number</span> <span m="522250">one,</span> <span m="522590">box</span> <span m="522960">number</span> <span m="523169">two,</span> <span m="523320">and</span> <span m="523409">box</span> <span m="523710">number</span> <span m="523890">three.</span> </p>
<p><span m="525260">Now,</span> <span m="525840">because</span> <span m="526320">people</span> <span m="526600">have</span> <span m="526690">seen</span> <span m="526960">the</span> <span m="527040">game</span> <span m="527310">show</span> <span m="527520">before,</span> <span m="528560">they</span> <span m="528740">knew</span> <span m="529080">that</span> <span m="529190">behind</span> <span m="529650">one</span> <span m="529890">of</span> <span m="529940">the</span> <span m="530050">doors</span> <span m="530620">was</span> <span m="530780">a</span> <span m="530840">brand</span> <span m="531200">new</span> <span m="531370">car</span> <span m="532340">or</span> <span m="532950">underneath</span> <span m="533410">one</span> <span m="533590">of</span> <span m="533630">the</span> <span m="533700">boxes</span> <span m="534380">was</span> <span m="534590">a</span> <span m="534690">diamond</span> <span m="535120">ring.</span> <span m="536520">And</span> <span m="536780">behind</span> <span m="537120">the</span> <span m="537280">other</span> <span m="537490">doors</span> <span m="539010">were</span> <span m="539170">goats</span> <span m="539800">or</span> <span m="539920">donkeys</span> <span m="540520">or</span> <span m="540670">something</span> <span m="541010">totally</span> <span m="541430">useless.</span> <span m="541960">So</span> <span m="542100">one</span> <span m="542400">of</span> <span m="542450">the</span> <span m="542550">three</span> <span m="542770">doors</span> <span m="543090">has</span> <span m="543230">a</span> <span m="543300">big</span> <span m="543550">prize,</span> <span m="544010">the</span> <span m="544110">other</span> <span m="544280">two</span> <span m="545080">got</span> <span m="545480">nothing.</span> <span m="547930">But</span> <span m="548580">because</span> <span m="549080">anybody</span> <span m="549490">would</span> <span m="549630">pay</span> <span m="549780">$100</span> <span m="550500">for</span> <span m="550740">a</span> <span m="551000">one-in-three</span> <span m="551470">chance</span> <span m="551790">at the</span> <span m="551940">car,</span> <span m="552410">they</span> <span m="552530">played.</span> <span m="552990">And</span> <span m="553210">they're</span> <span m="553310">on</span> <span m="553570">national</span> <span m="553950">TV.</span> <span m="554820">So</span> <span m="554940">they</span> <span m="555030">give</span> <span m="555200">the</span> <span m="555300">$100</span> <span m="555860">back</span> <span m="556130">to</span> <span m="556220">Monty,</span> <span m="557110">and</span> <span m="557710">they</span> <span m="557850">pick</span> <span m="558140">a</span> <span m="558230">door,</span> <span m="558930">say</span> <span m="559250">door</span> <span m="559480">number</span> <span m="559740">one.</span> </p>
<p><span m="561830">Monty</span> <span m="562220">wouldn't</span> <span m="562560">open</span> <span m="562930">door</span> <span m="563160">number</span> <span m="563360">one</span> <span m="563600">right</span> <span m="563840">away.</span> <span m="564690">What</span> <span m="564860">he</span> <span m="564920">would</span> <span m="565090">do</span> <span m="565300">is</span> <span m="565420">say,</span> <span m="566640">OK,</span> <span m="566980">you're</span> <span m="567240">sure</span> <span m="567430">you</span> <span m="567540">like</span> <span m="567780">that?</span> <span m="568110">All</span> <span m="568240">right.</span> <span m="569810">Carol,</span> <span m="570160">can</span> <span m="570300">you</span> <span m="570410">please</span> <span m="570700">reveal</span> <span m="571050">what's</span> <span m="571250">behind</span> <span m="571650">door</span> <span m="571870">number</span> <span m="572130">three?</span> <span m="573490">So</span> <span m="573700">Carol</span> <span m="574110">opens</span> <span m="574360">up</span> <span m="574510">door</span> <span m="574690">number</span> <span m="574910">three,</span> <span m="575270">and</span> <span m="575360">there's</span> <span m="575630">one</span> <span m="575820">of</span> <span m="575860">the</span> <span m="575950">donkeys</span> <span m="576360">sitting</span> <span m="576770">there</span> <span m="577150">or a</span> <span m="577470">goat sitting</span> <span m="577780">there.</span> <span m="578920">Well,</span> <span m="579060">now</span> <span m="579260">there's</span> <span m="579430">only</span> <span m="579610">two</span> <span m="579780">doors</span> <span m="580130">left.</span> <span m="580650">One</span> <span m="580920">of</span> <span m="580970">them</span> <span m="581080">has</span> <span m="581320">the</span> <span m="581420">car.</span> <span m="582500">So</span> <span m="582670">he'd</span> <span m="582850">ask</span> <span m="583100">the</span> <span m="583190">contestant,</span> <span m="584090">well,</span> <span m="584260">you've</span> <span m="584380">seen</span> <span m="584680">the</span> <span m="584770">goat.</span> <span m="586070">There's</span> <span m="586240">a</span> <span m="586280">car</span> <span m="586600">behind</span> <span m="586930">one</span> <span m="587070">of</span> <span m="587120">the</span> <span m="587210">two</span> <span m="587380">doors.</span> <span m="588510">Do</span> <span m="588620">you want</span> <span m="588750">to</span> <span m="588810">stick</span> <span m="589180">with</span> <span m="589300">door</span> <span m="589470">number</span> <span m="589690">one?</span> <span m="590620">Or</span> <span m="590790">do</span> <span m="590830">you</span> <span m="590910">want</span> <span m="591060">to</span> <span m="591120">switch</span> <span m="592230">and</span> <span m="592390">get</span> <span m="592570">door</span> <span m="592750">number</span> <span m="593000">two?</span> </p>
<p><span m="594260">And</span> <span m="594560">then</span> <span m="594750">the</span> <span m="594890">audience</span> <span m="595320">is</span> <span m="595420">screaming,</span> <span m="596220">you know,</span> <span m="596490">switch,</span> <span m="596960">stick,</span> <span m="597590">you know,</span> <span m="598170">door</span> <span m="598450">number one,</span> <span m="598660">door</span> <span m="598820">number</span> <span m="599060">two.</span> <span m="599630">You know,</span> <span m="600100">the</span> <span m="600220">guy's</span> <span m="600490">wife's</span> <span m="600790">there</span> <span m="601000">screaming</span> <span m="601320">at him and</span> <span m="601750">everything.</span> <span m="602840">You</span> <span m="602950">know,</span> <span m="603290">he's</span> <span m="603540">got</span> <span m="603710">to</span> <span m="604870">pick,</span> <span m="605420">so</span> <span m="605540">he</span> <span m="606120">picks</span> <span m="606390">one.</span> <span m="607300">And</span> <span m="608310">you</span> <span m="608450">know,</span> <span m="608650">if</span> <span m="608740">he</span> <span m="609570">picks</span> <span m="610170">the</span> <span m="610250">car,</span> <span m="611580">well,</span> <span m="611800">his</span> <span m="611940">wife's</span> <span m="612250">thrilled.</span> <span m="612620">They</span> <span m="612700">live</span> <span m="612820">happily</span> <span m="613210">ever.</span> <span m="613760">He's</span> <span m="613970">a</span> <span m="614370">hero on</span> <span m="614420">national</span> <span m="614790">TV.</span> <span m="615650">And</span> <span m="615810">if he</span> <span m="615990">picks</span> <span m="616230">the</span> <span m="616350">goat,</span> <span m="616610">you know,</span> <span m="616820">he's</span> <span m="617180">looking</span> <span m="617340">at</span> <span m="617470">a</span> <span m="617580">divorce</span> <span m="618100">and</span> <span m="618600">his</span> <span m="618730">career's</span> <span m="619110">ruined.</span> <span m="620560">So</span> <span m="620700">it's</span> <span m="620800">pretty</span> <span m="621010">important</span> <span m="621490">that he</span> <span m="621730">gets</span> <span m="622000">the</span> <span m="622100">right</span> <span m="622440">strategy</span> <span m="622980">here.</span> </p>
<p><span m="624250">Now,</span> <span m="624770">the</span> <span m="624860">question</span> <span m="625220">we're</span> <span m="625350">going</span> <span m="625450">to</span> <span m="625510">look</span> <span m="625730">at</span> <span m="625970">today</span> <span m="627380">is</span> <span m="627810">whether</span> <span m="628240">or</span> <span m="628270">not</span> <span m="628660">the</span> <span m="628730">contestant</span> <span m="629420">should</span> <span m="629670">switch,</span> <span m="630820">playing</span> <span m="631320">this</span> <span m="631560">game.</span> <span m="633050">Now,</span> <span m="633230">before</span> <span m="633640">we</span> <span m="633770">do</span> <span m="633970">the</span> <span m="634130">analysis,</span> <span m="635510">we're</span> <span m="635720">going</span> <span m="635820">to</span> <span m="635880">actually</span> <span m="636350">play</span> <span m="636690">the</span> <span m="636810">game</span> <span m="637320">in</span> <span m="637430">class.</span> <span m="638540">I've</span> <span m="638650">got</span> <span m="639050">three</span> <span m="639330">boxes</span> <span m="639830">here.</span> <span m="641640">The</span> <span m="641780">game</span> <span m="642010">we're</span> <span m="642100">going</span> <span m="642200">to</span> <span m="642260">play is</span> <span m="642570">the</span> <span m="642650">same</span> <span m="642920">idea.</span> <span m="643330">The</span> <span m="643450">prizes</span> <span m="644060">are</span> <span m="644140">a</span> <span m="644180">little</span> <span m="644390">different.</span> <span m="645300">Instead</span> <span m="645650">of</span> <span m="645720">a</span> <span m="645800">diamond</span> <span m="646220">ring</span> <span m="646490">or a</span> <span m="646600">brand</span> <span m="646910">new</span> <span m="647060">car,</span> <span m="647470">we've</span> <span m="647650">got</span> <span m="648470">$10</span> <span m="649510">Toski's</span> <span m="650040">gift</span> <span m="650220">certificates.</span> <span m="651700">You</span> <span m="651780">know,</span> <span m="652340">your</span> <span m="652500">tuition</span> <span m="652880">only</span> <span m="653040">goes</span> <span m="653170">so</span> <span m="653400">far</span> <span m="653670">here.</span> </p>
<p><span m="655370">Instead</span> <span m="655670">of</span> <span m="655800">donkeys</span> <span m="656270">or</span> <span m="656340">goats,</span> <span m="656690">we've</span> <span m="656790">got</span> <span m="656960">something</span> <span m="657180">much</span> <span m="657550">better,</span> <span m="658110">the</span> <span m="658230">ever-popular</span> <span m="658970">nerd pride</span> <span m="659690">pocket</span> <span m="660030">protector.</span> <span m="663470">And</span> <span m="663700">just to be</span> <span m="664030">clear,</span> <span m="665630">this</span> <span m="665830">is</span> <span m="665950">not</span> <span m="666230">the</span> <span m="666310">prize,</span> <span m="667160">the</span> <span m="667560">gift</span> <span m="667730">certificate</span> <span m="668230">here--</span> <span m="668590">this</span> <span m="668770">is</span> <span m="668880">much,</span> <span m="669310">much</span> <span m="669520">better.</span> </p>
<p><span m="670876">[AUDIENCE LAUGHTER]</span> </p>
<p><span m="671350">Now,</span> <span m="673350">in</span> <span m="673480">terms</span> <span m="673870">of</span> <span m="674540">Monty</span> <span m="674920">Hall,</span> <span m="675230">we</span> <span m="675330">don't</span> <span m="675500">have</span> <span m="675710">him,</span> <span m="675890">but</span> <span m="676040">you</span> <span m="676140">have</span> <span m="676410">me.</span> <span m="677680">But</span> <span m="677880">I</span> <span m="677940">am</span> <span m="678220">very</span> <span m="678490">happy</span> <span m="678790">to</span> <span m="678860">report</span> <span m="679270">that</span> <span m="679710">we're</span> <span m="679840">bringing</span> <span m="680160">Carol</span> <span m="680580">Merrill</span> <span m="681020">out</span> <span m="681200">of</span> <span m="681270">retirement.</span> <span m="682530">She's</span> <span m="682730">coming</span> <span m="683040">and</span> <span m="683100">making</span> <span m="683380">a</span> <span m="683430">special</span> <span m="683890">appearance</span> <span m="684450">today,</span> <span m="685630">I</span> <span m="685760">hope,</span> <span m="686390">at</span> <span m="686580">6042.</span> <span m="688680">Do</span> <span m="688770">we</span> <span m="688870">have</span> <span m="689050">Carol?</span> </p>
<p><span m="691036">GUEST SPEAKER: Hi, Tom.</span> </p>
<p><span m="691840">PROFESSOR: Oh,</span> <span m="692200">yes,</span> <span m="692550">we</span> <span m="692710">have</span> <span m="692980">Carol.</span> <span m="694140">That's</span> <span m="694440">scary.</span> </p>
<p><span m="696186">[AUDIENCE LAUGHTER]</span> </p>
<p><span m="697530">Oh,</span> <span m="697790">Jesus.</span> <span m="701160">OK.</span> </p>
<p><span m="702190">[APPLAUSE]</span> </p>
<p><span m="705340">As</span> <span m="705630">you</span> <span m="705750">can</span> <span m="705900">see,</span> <span m="706080">the</span> <span m="706210">years</span> <span m="706490">have</span> <span m="706610">not</span> <span m="706890">been</span> <span m="707090">kind</span> <span m="707470">to</span> <span m="707540">Carol.</span> <span m="709290">And</span> <span m="709830">your</span> <span m="710040">tuition</span> <span m="710480">only</span> <span m="710710">does</span> <span m="710990">goes</span> <span m="711200">so</span> <span m="711450">far.</span> <span m="712720">Now,</span> <span m="713090">we</span> <span m="713230">need</span> <span m="713860">three</span> <span m="714230">volunteers.</span> <span m="715840">We're</span> <span m="715940">going</span> <span m="716020">to</span> <span m="716070">play</span> <span m="716240">the</span> <span m="716330">game</span> <span m="716550">three</span> <span m="716880">times.</span> <span m="717760">All right,</span> <span m="718030">got</span> <span m="718210">a</span> <span m="718260">couple</span> <span m="718550">there.</span> <span m="719600">Why</span> <span m="719720">don't</span> <span m="719860">you</span> <span m="719970">come</span> <span m="720190">down?</span> <span m="720690">We've</span> <span m="720750">got</span> <span m="720910">three</span> <span m="721080">volunteers</span> <span m="721760">here.</span> <span m="723180">All</span> <span m="723320">right,</span> <span m="723660">we'll</span> <span m="724690">get</span> <span m="724880">the</span> <span m="725130">boxes</span> <span m="725600">set</span> <span m="725820">up.</span> <span m="728490">And</span> <span m="728580">Carol,</span> <span m="728890">why don't</span> <span m="729180">you</span> <span m="729310">set</span> <span m="729600">those</span> <span m="729870">up</span> <span m="730090">and</span> <span m="730820">load</span> <span m="731060">them</span> <span m="731220">up</span> <span m="731430">with--</span> <span m="733440">these</span> <span m="733680">are</span> <span m="733730">the</span> <span m="733820">gift</span> <span m="734060">certificates.</span> <span m="734660">That's</span> <span m="734950">the</span> <span m="735520">pocket</span> <span m="735860">protector</span> <span m="736280">there.</span> <span m="737150">All right, now,</span> <span m="737370">don't</span> <span m="737630">look.</span> <span m="737770">You've</span> <span m="737850">got</span> <span m="737980">to</span> <span m="738030">get</span> <span m="738170">back</span> <span m="738410">over</span> <span m="738560">there</span> <span m="738870">so</span> <span m="738970">you</span> <span m="739070">can't</span> <span m="739190">see--</span> </p>
<p><span m="739590">[AUDIENCE LAUGHTER]</span> </p>
<p><span m="740550">And</span> <span m="740870">no,</span> <span m="741410">no,</span> <span m="741790">no,</span> <span m="742150">now,</span> <span m="742510">you've</span> <span m="742580">got</span> <span m="742740">the</span> <span m="742830">film</span> <span m="743140">going,</span> <span m="743400">you know,</span> <span m="743730">here</span> <span m="744205">we go.</span> <span m="745070">All</span> <span m="745460">right.</span> <span m="746910">So</span> <span m="747930">Carol</span> <span m="748300">is</span> <span m="748380">going</span> <span m="748490">to</span> <span m="748540">put</span> <span m="748700">the</span> <span m="748780">prize</span> <span m="749260">in</span> <span m="749660">one</span> <span m="749880">of</span> <span m="749920">the</span> <span m="750000">boxes</span> <span m="750810">and</span> <span m="751250">the</span> <span m="751330">pocket</span> <span m="751680">protectors</span> <span m="752200">in</span> <span m="752280">the</span> <span m="752450">other.</span> <span m="755210">Now,</span> <span m="755310">I</span> <span m="755410">want</span> <span m="755520">you</span> <span m="755690">all</span> <span m="755870">to be</span> <span m="756020">thinking,</span> <span m="757340">you</span> <span m="757420">know,</span> <span m="757560">what's</span> <span m="757870">the</span> <span m="757980">right</span> <span m="758200">strategy</span> <span m="758850">here?</span> <span m="759270">Is</span> <span m="759440">it</span> <span m="759540">better</span> <span m="759800">to</span> <span m="759880">switch?</span> <span m="761020">Does</span> <span m="761230">it</span> <span m="761370">matter?</span> <span m="763030">What</span> <span m="763260">are</span> <span m="763290">the</span> <span m="763400">chances</span> <span m="763840">of</span> <span m="763920">winning?</span> <span m="764960">One</span> <span m="765180">in</span> <span m="765290">three?</span> <span m="765840">One</span> <span m="766070">in two?</span> </p>
<p><span m="767570">All</span> <span m="767830">right,</span> <span m="768740">so</span> <span m="769020">who's</span> <span m="769230">our</span> <span m="769330">first</span> <span m="769610">volunteer</span> <span m="770180">here?</span> <span m="771930">All right,</span> <span m="772160">come</span> <span m="772350">on</span> <span m="772490">over.</span> <span m="772950">We're</span> <span m="773260">going</span> <span m="773340">to</span> <span m="773400">write</span> <span m="773570">your</span> <span m="773660">name</span> <span m="773870">on</span> <span m="773920">the</span> <span m="774000">board</span> <span m="774250">because we're</span> <span m="774340">going</span> <span m="774450">to</span> <span m="774540">keep</span> <span m="774760">track</span> <span m="775010">of</span> <span m="775100">what</span> <span m="775290">happens</span> <span m="775730">here.</span> <span m="777590">What's</span> <span m="778070">your</span> <span m="778200">name?</span> </p>
<p><span m="779266">TERRANCE: Terrance.</span> </p>
<p><span m="779670">PROFESSOR: Terrance.</span> <span m="782570">All</span> <span m="782800">right.</span> <span m="783530">Now,</span> <span m="784670">first</span> <span m="784900">thing</span> <span m="784990">you've</span> <span m="785100">got</span> <span m="785250">to</span> <span m="785340">do,</span> <span m="785960">Terrance,</span> <span m="786590">is</span> <span m="787410">pick</span> <span m="787830">a</span> <span m="787890">box</span> <span m="789160">as</span> <span m="789420">your</span> <span m="789610">basic</span> <span m="790070">choice</span> <span m="790740">to</span> <span m="790830">start.</span> </p>
<p><span m="791980">TERRANCE: Two.</span> </p>
<p><span m="792610">PROFESSOR: Number</span> <span m="792930">two,</span> <span m="793650">OK.</span> <span m="796260">All</span> <span m="796520">right,</span> <span m="796880">you</span> <span m="797050">picked</span> <span m="797910">number</span> <span m="798230">two.</span> <span m="800106">Hm,</span> <span m="800990">interesting</span> <span m="801440">choice.</span> <span m="803070">Now,</span> <span m="804200">Carol,</span> <span m="804520">can</span> <span m="804660">you</span> <span m="805060">show</span> <span m="805430">him</span> <span m="805740">one</span> <span m="805920">of</span> <span m="805990">the</span> <span m="806090">other</span> <span m="806270">boxes?</span> <span m="814620">All right,</span> <span m="814880">which on</span> <span m="815230">was</span> <span m="815360">that</span> <span m="815540">you</span> <span m="815630">showed</span> <span m="816060">there?</span> <span m="816260">You</span> <span m="816380">revealed</span> <span m="817060">number</span> <span m="817480">three.</span> <span m="820990">And</span> <span m="821190">what</span> <span m="821410">was</span> <span m="821580">under</span> <span m="821950">number</span> <span m="822240">three?</span> </p>
<p><span m="823832">AUDIENCE: Pocket protector.</span> </p>
<p><span m="825275">PROFESSOR: Pocket protector.</span> <span m="826718">Oh, God,</span> <span m="828170">you</span> <span m="828290">may</span> <span m="828460">never</span> <span m="828720">live</span> <span m="828890">this</span> <span m="829060">down,</span> <span m="829320">Carol.</span> <span m="830380">OK.</span> <span m="833820">Now,</span> <span m="834100">Terrance,</span> <span m="834690">as you know,</span> <span m="835881">one</span> <span m="836280">of</span> <span m="836370">these</span> <span m="836740">has</span> <span m="837670">the</span> <span m="837760">gift</span> <span m="837970">certificate.</span> <span m="839280">One</span> <span m="839590">of</span> <span m="839680">them</span> <span m="840050">has</span> <span m="840360">another</span> <span m="840690">pocket</span> <span m="841070">protector.</span> <span m="842300">And</span> <span m="842460">now,</span> <span m="842580">you've</span> <span m="842770">got</span> <span m="842930">to</span> <span m="842990">decide,</span> <span m="844590">are</span> <span m="844730">you</span> <span m="844860">going</span> <span m="844980">to</span> <span m="845050">stick</span> <span m="845530">with</span> <span m="845990">number</span> <span m="846300">two,</span> <span m="847050">or</span> <span m="847310">are</span> <span m="847480">you</span> <span m="847610">going</span> <span m="847740">to</span> <span m="847820">switch?</span> <span m="848470">In</span> <span m="848560">fact,</span> <span m="848830">class,</span> <span m="849710">you know,</span> <span m="849970">Terrance</span> <span m="850300">may</span> <span m="850450">need</span> <span m="850660">some</span> <span m="850800">help</span> <span m="851070">here,</span> <span m="851250">figuring</span> <span m="851540">this</span> <span m="851780">out,</span> <span m="852030">so</span> <span m="852140">what</span> <span m="852270">do</span> <span m="852310">you</span> <span m="852420">recommend?</span> </p>
<p><span m="853420">[AUDIENCE  CHATTER]</span> </p>
<p><span m="859180">TERRANCE: I'll switch.</span> </p>
<p><span m="860630">PROFESSOR: He</span> <span m="860840">is</span> <span m="861000">going</span> <span m="861330">to</span> <span m="861490">switch.</span> <span m="862960">All</span> <span m="863160">right,</span> <span m="863610">and</span> <span m="863940">Carol,</span> <span m="864440">can</span> <span m="864630">you</span> <span m="864840">reveal</span> <span m="865310">his</span> <span m="865570">prize?</span> <span m="869670">What</span> <span m="869710">did you</span> <span m="869800">get?</span> <span m="870710">Ah,</span> <span m="871620">he gets</span> <span m="872070">the</span> <span m="872180">prize.</span> <span m="873000">So</span> <span m="873230">he</span> <span m="873400">wins.</span> <span m="875320">Very</span> <span m="875660">nice,</span> <span m="876150">OK.</span> <span m="876650">So</span> <span m="876770">you</span> <span m="876850">get</span> <span m="877030">$10</span> <span m="877320">at</span> <span m="877610">Tosci's.</span> <span m="878200">Well</span> <span m="878470">done.</span> </p>
<p><span m="879570">Who's</span> <span m="879820">next?</span> <span m="880960">What's</span> <span m="881100">your</span> <span m="881200">name?</span> </p>
<p><span m="882050">CHINUA: Chinua.</span> </p>
<p><span m="882530">PROFESSOR: Chinua.</span> <span m="883120">How</span> <span m="883350">do</span> <span m="883450">I</span> <span m="883540">spell</span> <span m="883870">that?</span> </p>
<p><span m="884578">CHINUA: Chin-U-A.</span> </p>
<p><span m="886670">PROFESSOR: All</span> <span m="886820">right,</span> <span m="887120">easy.</span> <span m="887500">OK.</span> <span m="888460">Now,</span> <span m="888760">Chinua,</span> <span m="889190">which--</span> <span m="889620">have we</span> <span m="889710">got</span> <span m="889820">new</span> <span m="890030">prizes</span> <span m="890570">loaded</span> <span m="890860">in</span> <span m="890970">there?</span> <span m="891660">All</span> <span m="891920">right.</span> <span m="892690">Now,</span> <span m="892850">it</span> <span m="893000">may</span> <span m="893150">not</span> <span m="893330">be</span> <span m="893400">the</span> <span m="893490">same</span> <span m="893740">as</span> <span m="893830">last</span> <span m="894160">time.</span> <span m="895360">OK.</span> <span m="895730">What</span> <span m="895950">box</span> <span m="896220">do</span> <span m="896290">you</span> <span m="896370">like?</span> </p>
<p><span m="896830">CHINUA: Number</span> <span m="896970">three.</span> </p>
<p><span m="898060">PROFESSOR: Number</span> <span m="898550">three.</span> <span m="899410">Which</span> <span m="899580">one</span> <span m="899730">is</span> <span m="899820">number</span> <span m="900060">three?</span> <span m="900420">Is that</span> <span m="900650">the</span> <span m="900730">one</span> <span m="900900">close</span> <span m="901260">to</span> <span m="901350">me?</span> <span m="901620">Yeah,</span> <span m="901730">that</span> <span m="901900">is.</span> <span m="902240">OK.</span> <span m="902620">Number</span> <span m="902820">three.</span> <span m="903840">All right,</span> <span m="904100">Carol,</span> <span m="904410">can</span> <span m="904560">you</span> <span m="904690">show</span> <span m="904990">him</span> <span m="905220">one</span> <span m="905390">of</span> <span m="905440">the</span> <span m="905540">other</span> <span m="905690">boxes,</span> <span m="906160">please?</span> </p>
<p><span m="913550">Carol</span> <span m="913950">shows</span> <span m="914260">you</span> <span m="914380">number</span> <span m="914710">two.</span> <span m="916820">All</span> <span m="917250">right,</span> <span m="917480">Chinua,</span> <span m="917800">what</span> <span m="918070">do</span> <span m="918110">you</span> <span m="918250">think?</span> <span m="919110">Do</span> <span m="919380">you</span> <span m="919530">want</span> <span m="919690">to</span> <span m="919750">stick</span> <span m="920700">or</span> <span m="920870">switch?</span> <span m="922780">Do you</span> <span m="922950">want to</span> <span m="923010">stick</span> <span m="923350">with</span> <span m="923500">three</span> <span m="924000">or</span> <span m="924200">switch</span> <span m="924600">to</span> <span m="924730">one?</span> <span m="925240">One</span> <span m="925540">had</span> <span m="925740">it</span> <span m="925910">last</span> <span m="926300">time,</span> <span m="926580">right?</span> </p>
<p><span m="928203">[AUDIENCE CHATTER]</span> </p>
<p><span m="932532">CHINUA: Switch.</span> </p>
<p><span m="933020">PROFESSOR: He's</span> <span m="933200">going</span> <span m="933570">to</span> <span m="933670">switch.</span> <span m="935246">OK,</span> <span m="936160">so</span> <span m="936590">that</span> <span m="936820">means</span> <span m="937070">you</span> <span m="937200">are</span> <span m="937430">picking</span> <span m="937790">number</span> <span m="938120">one.</span> <span m="938700">Carol,</span> <span m="939050">can</span> <span m="939180">you</span> <span m="939280">show</span> <span m="939610">him</span> <span m="939830">what</span> <span m="939990">he's</span> <span m="940280">won?</span> <span m="942130">Another</span> <span m="942580">gift</span> <span m="942950">certificate.</span> <span m="945416">All right,</span> <span m="945870">now,</span> <span m="946230">Carol</span> <span m="946540">it is</span> <span m="946680">OK</span> <span m="946950">to</span> <span m="947010">use</span> <span m="947190">a</span> <span m="947260">different</span> <span m="947550">box</span> <span m="947950">for the</span> <span m="948120">gift</span> <span m="948310">certificate.</span> </p>
<p><span m="948960">[LAUGHTER]</span> </p>
<p><span m="949800">We</span> <span m="950120">can</span> <span m="950440">move them</span> <span m="950780">around.</span> <span m="952170">All</span> <span m="952390">right,</span> <span m="952610">who's our</span> <span m="952730">last</span> <span m="953020">contestant?</span> <span m="953970">What's</span> <span m="954130">your</span> <span m="954220">name?</span> </p>
<p><span m="955110">TY: Ty.</span> </p>
<p><span m="955600">PROFESSOR: T-Y,</span> <span m="955920">OK.</span> <span m="956736">That's</span> <span m="957232">good.</span> <span m="959220">Oh,</span> <span m="959530">yeah,</span> <span m="959600">yeah,</span> <span m="959850">turn around.</span> <span m="960970">No peeking</span> <span m="961300">here.</span> <span m="966600">OK.</span> <span m="967250">Ty,</span> <span m="967690">what's</span> <span m="967960">your</span> <span m="968130">first</span> <span m="968480">choice?</span> </p>
<p><span m="970352">TY: One.</span> </p>
<p><span m="970850">PROFESSOR: Box</span> <span m="971350">number</span> <span m="971660">one.</span> <span m="971945">Yeah,</span> <span m="972230">that's</span> <span m="972510">been</span> <span m="972640">the</span> <span m="972730">one</span> <span m="972980">with</span> <span m="973210">the</span> <span m="973800">gift</span> <span m="973970">certificate,</span> <span m="974520">for</span> <span m="974620">sure.</span> <span m="975500">Carol,</span> <span m="977254">can you show him</span> <span m="978570">another</span> <span m="979330">box?</span> <span m="980500">Way</span> <span m="980730">too</span> <span m="980890">many</span> <span m="981090">cameras</span> <span m="981530">in</span> <span m="981580">this</span> <span m="981770">room.</span> <span m="982860">This</span> <span m="983000">is</span> <span m="983140">not</span> <span m="983410">good</span> <span m="983620">for</span> <span m="983730">Carol.</span> </p>
<p><span m="984982">[LAUGHTER]</span> </p>
<p><span m="986670">OK,</span> <span m="987260">Carol</span> <span m="987590">has</span> <span m="987720">revealed</span> <span m="988550">box</span> <span m="988940">number</span> <span m="989230">two.</span> </p>
<p><span m="990830">[AUDIENCE CHATTER]</span> </p>
<p><span m="991630">Switch</span> <span m="991930">has</span> <span m="992070">worked</span> <span m="992500">twice in a</span> <span m="992910">row.</span> </p>
<p><span m="994050">AUDIENCE: Now</span> <span m="994540">you gotta stay.</span> </p>
<p><span m="995030">AUDIENCE: Take them</span> <span m="995260">both.</span> </p>
<p><span m="996890">PROFESSOR: Just</span> <span m="997270">one.</span> </p>
<p><span m="1001798">AUDIENCE: Well, if you</span> <span m="1002269">switch,</span> <span m="1002740">there's</span> <span m="1003040">two-thirds</span> <span m="1003526">chance of winning,</span> <span m="1004012">and they</span> <span m="1004500">already</span> <span m="1004860">won so--</span> </p>
<p><span m="1006288">[LAUGHTER]</span> </p>
<p><span m="1007720">PROFESSOR: Oh,</span> <span m="1008040">getting</span> <span m="1008360">some</span> <span m="1008550">theory</span> <span m="1008640">here</span> <span m="1008980">now.</span> <span m="1010660">What</span> <span m="1010965">do</span> <span m="1011270">you think,</span> <span m="1011690">Ty?</span> <span m="1013210">Switch.</span> <span m="1014520">All</span> <span m="1014860">right.</span> <span m="1015190">What</span> <span m="1015350">happened?</span> <span m="1018360">What do you</span> <span m="1018610">know?</span> <span m="1020190">All right,</span> <span m="1020480">now,</span> <span m="1020760">gentlemen,</span> <span m="1021260">I'd like</span> <span m="1021600">you to</span> <span m="1021820">explain</span> <span m="1022650">why--</span> <span m="1023330">each</span> <span m="1023770">in turn,</span> <span m="1024000">why</span> <span m="1024180">did</span> <span m="1024329">you</span> <span m="1024440">switch?</span> </p>
<p><span m="1026515">TERRANCE: I switched--</span> <span m="1027010">it seemed</span> <span m="1027505">like the</span> <span m="1028000">majority vote.</span> </p>
<p><span m="1028990">PROFESSOR: The majority</span> <span m="1029485">knew what</span> <span m="1029980">they were</span> <span m="1030475">talking about,</span> <span m="1031465">OK.</span> <span m="1033940">You have faith</span> <span m="1034435">in these guys, huh?</span> <span m="1035425">All right.</span> <span m="1036910">How about you,</span> <span m="1037420">Chinua,</span> <span m="1037589">why</span> <span m="1037770">did you</span> <span m="1038140">switch?</span> </p>
<p><span m="1038976">CHINUA: Um, I was</span> <span m="1039442">told</span> <span m="1039908">that</span> <span m="1040374">it's more</span> <span m="1040840">likely to be</span> <span m="1041306">if you</span> <span m="1041772">switch.</span> </p>
<p><span m="1042704">PROFESSOR: And you</span> <span m="1043170">believe</span> <span m="1043480">what you hear,</span> <span m="1043789">huh?</span> </p>
<p><span m="1044576">[LAUGHTER]</span> </p>
<p></p>
<p><span m="1048392">TY: I don't</span> <span m="1048869">know who said it, but</span> <span m="1049346">it sounded</span> <span m="1049823">good.</span> </p>
<p><span m="1051254">PROFESSOR: Sounded good</span> <span m="1051731">Yeah, that two-thirds thing.</span> <span m="1052535">Well, as you can see,</span> <span m="1052850">you're</span> <span m="1053030">guaranteed</span> <span m="1053530">to</span> <span m="1053610">win</span> <span m="1053810">if</span> <span m="1053910">you</span> <span m="1054000">switch,</span> <span m="1054530">right?</span> </p>
<p><span m="1055834">[LAUGHTER]</span> </p>
<p><span m="1056286">All right,</span> <span m="1056740">Thank</span> <span m="1057050">you</span> <span m="1057150">very</span> <span m="1057410">much.</span> <span m="1057830">Enjoy</span> <span m="1058060">the</span> <span m="1058230">ice</span> <span m="1058450">cream.</span> <span m="1059860">OK.</span> <span m="1060460">So</span> <span m="1060940">how</span> <span m="1061200">many</span> <span m="1061500">people</span> <span m="1063080">think</span> <span m="1063480">it</span> <span m="1063790">helps</span> <span m="1064290">to</span> <span m="1064390">switch?</span> <span m="1065340">Raise</span> <span m="1065580">your</span> <span m="1065690">hand</span> <span m="1065890">if</span> <span m="1065970">you</span> <span m="1066090">think</span> <span m="1066340">it</span> <span m="1066430">helps</span> <span m="1066670">to</span> <span m="1066740">switch.</span> <span m="1067690">How</span> <span m="1067840">many</span> <span m="1068030">people</span> <span m="1068340">think</span> <span m="1068530">it</span> <span m="1068650">does</span> <span m="1068850">not</span> <span m="1069200">help</span> <span m="1069910">to</span> <span m="1070020">switch?</span> <span m="1070400">Raise</span> <span m="1070610">your</span> <span m="1070740">hand.</span> <span m="1071960">Anybody?</span> <span m="1072530">Oh, everybody</span> <span m="1072820">likes</span> <span m="1073110">the</span> <span m="1073210">switch</span> <span m="1073580">thing</span> <span m="1073800">here.</span> <span m="1075860">Does</span> <span m="1076040">this</span> <span m="1076290">data</span> <span m="1076580">help</span> <span m="1076870">you</span> <span m="1076990">decide?</span> <span m="1079450">No.</span> <span m="1079960">Anything</span> <span m="1080270">could have</span> <span m="1080430">happened</span> <span m="1080760">with</span> <span m="1080900">three</span> <span m="1081080">trials.</span> <span m="1081610">In</span> <span m="1081710">fact,</span> <span m="1081790">we'll</span> <span m="1081940">talk</span> <span m="1082180">about</span> <span m="1082880">statistical</span> <span m="1083640">inference</span> <span m="1083895">and</span> <span m="1084260">statistical</span> <span m="1084780">sampling</span> <span m="1085310">later.</span> </p>
<p><span m="1088130">If</span> <span m="1088280">you</span> <span m="1088390">switch,</span> <span m="1088790">what</span> <span m="1088930">do</span> <span m="1088970">you</span> <span m="1089060">think</span> <span m="1089250">the</span> <span m="1089390">odds</span> <span m="1089620">of</span> <span m="1089710">winning</span> <span m="1090030">are?</span> </p>
<p><span m="1090783">AUDIENCE: 100%</span> </p>
<p><span m="1091650">PROFESSOR: 100%?</span> <span m="1092490">OK,</span> <span m="1093130">who likes</span> <span m="1093590">100%?</span> <span m="1094630">Raise</span> <span m="1094840">your</span> <span m="1094970">hand.</span> <span m="1095810">Well, we've</span> <span m="1096010">got</span> <span m="1096160">a</span> <span m="1096210">couple.</span> <span m="1097410">How</span> <span m="1097540">many</span> <span m="1097730">people</span> <span m="1098040">think</span> <span m="1098220">it's</span> <span m="1098370">50-50</span> <span m="1099510">if</span> <span m="1099650">you</span> <span m="1099740">switch?</span> <span m="1100940">Yeah,</span> <span m="1101250">a</span> <span m="1101270">lot.</span> <span m="1101810">How</span> <span m="1101960">many</span> <span m="1102160">people</span> <span m="1102470">think</span> <span m="1102950">it's</span> <span m="1103190">worse</span> <span m="1103460">than</span> <span m="1103590">50-50</span> <span m="1104690">if</span> <span m="1104810">you</span> <span m="1104920">switch?</span> <span m="1106250">Not</span> <span m="1106810">many.</span> <span m="1106990">How many people</span> <span m="1107250">think</span> <span m="1107400">it's</span> <span m="1107540">better</span> <span m="1107890">than</span> <span m="1108070">50-50?</span> <span m="1109426">Ah,</span> <span m="1109810">that's</span> <span m="1110610">I think</span> <span m="1111490">most</span> <span m="1111750">of</span> <span m="1111830">you</span> <span m="1111940">there,</span> <span m="1112210">pretty</span> <span m="1112440">close.</span> <span m="1113250">All</span> <span m="1113600">right,</span> <span m="1113810">now,</span> <span m="1114110">the</span> <span m="1114250">answers</span> <span m="1114610">to</span> <span m="1114670">these</span> <span m="1114910">questions</span> <span m="1116090">are</span> <span m="1116200">very</span> <span m="1116600">easy</span> <span m="1116800">to</span> <span m="1116890">work</span> <span m="1117120">out--</span> <span m="1117630">and</span> <span m="1117750">we're</span> <span m="1117880">going</span> <span m="1117990">to</span> <span m="1118070">do</span> <span m="1118240">it.</span> <span m="1118320">We're</span> <span m="1118430">going</span> <span m="1118530">to</span> <span m="1118590">figure</span> <span m="1118910">out</span> <span m="1118990">the</span> <span m="1119080">probability</span> <span m="1119600">exactly--</span> <span m="1120700">if</span> <span m="1120970">you</span> <span m="1121070">use</span> <span m="1121680">mathematics.</span> <span m="1122960">But</span> <span m="1123100">it's</span> <span m="1123260">really</span> <span m="1123700">tricky</span> <span m="1124110">if</span> <span m="1124210">you</span> <span m="1124310">try</span> <span m="1124500">to</span> <span m="1124580">use</span> <span m="1124810">intuition</span> <span m="1125460">or</span> <span m="1126020">hand</span> <span m="1126390">waving</span> <span m="1126930">arguments.</span> </p>
<p><span m="1128490">Now,</span> <span m="1128570">for</span> <span m="1128710">example,</span> <span m="1130080">this</span> <span m="1130330">game--</span> <span m="1131320">Carol</span> <span m="1131610">wants</span> <span m="1131820">to</span> <span m="1131920">leave--</span> <span m="1132230">yeah,</span> <span m="1132410">Carol,</span> <span m="1132950">let's</span> <span m="1133110">give</span> <span m="1133270">Carol</span> <span m="1133640">a hand</span> <span m="1133930">here.</span> <span m="1134220">She</span> <span m="1134810">did</span> <span m="1135030">a wonderful</span> <span m="1135460">job.</span> <span m="1136790">Well</span> <span m="1136990">done.</span> </p>
<p><span m="1137526">[APPLAUSE]</span> </p>
<p></p>
<p><span m="1140020">Yikes.</span> <span m="1142260">I'm</span> <span m="1142440">going</span> <span m="1142510">to</span> <span m="1142560">get</span> <span m="1142700">in</span> <span m="1142750">trouble</span> <span m="1142990">here,</span> <span m="1143180">too,</span> <span m="1143460">now.</span> <span m="1144040">You know?</span> <span m="1145590">OK.</span> <span m="1147770">For</span> <span m="1147910">example,</span> <span m="1148880">this</span> <span m="1149210">game,</span> <span m="1149565">and</span> <span m="1149920">this</span> <span m="1150170">particular</span> <span m="1150640">version</span> <span m="1151040">of</span> <span m="1151170">it,</span> <span m="1151710">was</span> <span m="1151980">the</span> <span m="1152090">subject</span> <span m="1152750">of</span> <span m="1153450">a</span> <span m="1153540">series</span> <span m="1154020">of</span> <span m="1154120">articles</span> <span m="1154580">in</span> <span m="1154680"><i>Parade</i></span> <span m="1155150"><i>Magazine</i>.</span> <span m="1156050">This</span> <span m="1156230">is</span> <span m="1156860">an</span> <span m="1156930">old</span> <span m="1157150">version</span> <span m="1157400">of</span> <span m="1157470"><i>Parade</i></span> <span m="1157830"><i>Magazine</i>.</span> <span m="1159500">And</span> <span m="1159750">what</span> <span m="1159830">I'm</span> <span m="1159930">going</span> <span m="1160010">to</span> <span m="1160050">do</span> <span m="1160180">is</span> <span m="1160300">hand</span> <span m="1160710">out</span> <span m="1160890">the</span> <span m="1160980">series</span> <span m="1161550">of</span> <span m="1161690">articles.</span> <span m="1162260">And</span> <span m="1162330">the</span> <span m="1162460">TAs</span> <span m="1162810">are</span> <span m="1162850">going</span> <span m="1162930">to</span> <span m="1162980">help</span> <span m="1163170">me</span> <span m="1163730">hand</span> <span m="1164000">these</span> <span m="1164220">out.</span> <span m="1166130">There</span> <span m="1166280">was</span> <span m="1166630">three</span> <span m="1167040">of</span> <span m="1167120">them</span> <span m="1167320">with a</span> <span m="1167410">lot</span> <span m="1167650">of</span> <span m="1167760">write-ins</span> <span m="1168290">here.</span> <span m="1168580">I'm going</span> <span m="1168710">to</span> <span m="1169100">give you</span> <span m="1169483">a bundle</span> <span m="1169866">here.</span> <span m="1172724">There</span> <span m="1173150">you</span> <span m="1173190">go.</span> <span m="1175340">There you go.</span> <span m="1175400">I'll give the</span> <span m="1175520">rest</span> <span m="1175880">to</span> <span m="1176240">Oscar.</span> <span m="1179980">Great.</span> </p>
<p><span m="1194100">Now,</span> <span m="1195190">Marilyn</span> <span m="1195650">has</span> <span m="1195870">a</span> <span m="1195940">column</span> <span m="1196700">every</span> <span m="1197000">week</span> <span m="1197880">in</span> <span m="1198060"><i>Parade.</i></span> <span m="1199030">And</span> <span m="1199210">people</span> <span m="1199540">write</span> <span m="1199840">in</span> <span m="1200140">questions,</span> <span m="1200760">usually</span> <span m="1201200">about</span> <span m="1201440">their</span> <span m="1201570">love</span> <span m="1201820">life</span> <span m="1202350">and</span> <span m="1203330">weird</span> <span m="1203680">things.</span> <span m="1204700">And</span> <span m="1204950">she</span> <span m="1205150">answers</span> <span m="1205530">them.</span> <span m="1206350">And</span> <span m="1206630">she's</span> <span m="1207020">well</span> <span m="1207260">qualified</span> <span m="1207730">to</span> <span m="1207800">do</span> <span m="1207990">this</span> <span m="1208210">because</span> <span m="1208690">she's</span> <span m="1208940">listed</span> <span m="1209260">in</span> <span m="1209330">the</span> <span m="1209430">Guinness</span> <span m="1209820">Book</span> <span m="1210100">of</span> <span m="1210190">World</span> <span m="1210470">Records</span> <span m="1210890">as</span> <span m="1211010">having</span> <span m="1211970">the</span> <span m="1212040">highest</span> <span m="1212530">IQ</span> <span m="1213000">of</span> <span m="1213110">a</span> <span m="1213160">human</span> <span m="1213430">being</span> <span m="1213740">ever,</span> <span m="1214430">at</span> <span m="1214820">228.</span> <span m="1216350">And</span> <span m="1216510">so,</span> <span m="1216990">you know</span> <span m="1217290">she's</span> <span m="1217550">in</span> <span m="1217640">some</span> <span m="1217860">sense</span> <span m="1218100">qualified</span> <span m="1218640">to</span> <span m="1218710">answer</span> <span m="1219090">any</span> <span m="1219320">question</span> <span m="1219760">that</span> <span m="1219870">is</span> <span m="1219980">written</span> <span m="1220290">in.</span> </p>
<p><span m="1221600">Now,</span> <span m="1222150">in</span> <span m="1222270">September</span> <span m="1222890">of</span> <span m="1223510">1990--</span> <span m="1225010">and</span> <span m="1225160">we're</span> <span m="1225270">passing</span> <span m="1225690">out</span> <span m="1225830">the</span> <span m="1225910">articles</span> <span m="1226390">now</span> <span m="1227760">and</span> <span m="1227980">it</span> <span m="1228080">shows</span> <span m="1228550">the</span> <span m="1229910">first</span> <span m="1230250">article--</span> <span m="1232880">this</span> <span m="1233160">guy</span> <span m="1233380">named</span> <span m="1233610">Craig</span> <span m="1233960">Whitaker</span> <span m="1234820">writes</span> <span m="1235210">in</span> <span m="1236080">and</span> <span m="1236300">asks</span> <span m="1236700">Marilyn</span> <span m="1237560">if</span> <span m="1237870">you</span> <span m="1238020">should</span> <span m="1238270">switch</span> <span m="1239050">in</span> <span m="1239180">this</span> <span m="1239380">game.</span> <span m="1240610">So</span> <span m="1240730">you</span> <span m="1240840">can</span> <span m="1240930">all</span> <span m="1241090">read</span> <span m="1241300">that.</span> <span m="1241510">It</span> <span m="1241630">says,</span> <span m="1241900">suppose</span> <span m="1242380">you're</span> <span m="1242520">on</span> <span m="1242620">a</span> <span m="1242660">game</span> <span m="1242980">show</span> <span m="1243350">and</span> <span m="1243450">you're</span> <span m="1243570">given</span> <span m="1243830">the</span> <span m="1243920">choice</span> <span m="1244250">of</span> <span m="1244360">three</span> <span m="1244610">doors.</span> <span m="1245500">Behind</span> <span m="1245820">one</span> <span m="1246100">door</span> <span m="1246310">is</span> <span m="1246400">a</span> <span m="1246450">car.</span> <span m="1247120">Behind</span> <span m="1247410">the</span> <span m="1247550">others,</span> <span m="1248130">goats.</span> <span m="1249180">You</span> <span m="1249380">pick</span> <span m="1249600">a</span> <span m="1249690">door,</span> <span m="1250090">say</span> <span m="1250330">number</span> <span m="1250600">one,</span> <span m="1251400">and</span> <span m="1251540">the</span> <span m="1251620">host,</span> <span m="1252470">who</span> <span m="1252600">knows</span> <span m="1252900">what's</span> <span m="1253060">behind</span> <span m="1253430">the</span> <span m="1253500">doors,</span> <span m="1254120">opens</span> <span m="1254470">another</span> <span m="1254820">one,</span> <span m="1255080">say</span> <span m="1255770">door</span> <span m="1256000">three,</span> <span m="1256860">which</span> <span m="1257240">has</span> <span m="1257520">a</span> <span m="1257600">goat.</span> <span m="1258910">He</span> <span m="1259050">says to you,</span> <span m="1259700">do</span> <span m="1259830">you</span> <span m="1259950">want</span> <span m="1260240">to</span> <span m="1260350">pick</span> <span m="1260650">door</span> <span m="1260870">number</span> <span m="1261140">two?</span> <span m="1262480">Is</span> <span m="1262750">it</span> <span m="1262890">to</span> <span m="1262950">your</span> <span m="1263130">advantage</span> <span m="1264570">to</span> <span m="1264720">switch?</span> </p>
<p><span m="1267720">Now,</span> <span m="1267920">Marilyn</span> <span m="1268390">writes</span> <span m="1268680">back</span> <span m="1269360">saying,</span> <span m="1269920">very</span> <span m="1270220">plainly,</span> <span m="1270670">yes,</span> <span m="1270930">you</span> <span m="1271030">should</span> <span m="1271250">switch.</span> <span m="1272070">The</span> <span m="1272200">first</span> <span m="1272580">door</span> <span m="1272830">had</span> <span m="1272990">a</span> <span m="1273060">one-in-three</span> <span m="1273550">chance</span> <span m="1273870">of</span> <span m="1273970">winning.</span> <span m="1274880">The</span> <span m="1274990">second</span> <span m="1275520">door</span> <span m="1275840">has</span> <span m="1276080">a</span> <span m="1276330">two-in-three</span> <span m="1277060">chance.</span> <span m="1278530">Here's</span> <span m="1278900">a</span> <span m="1278970">good</span> <span m="1279210">way</span> <span m="1279400">to</span> <span m="1279490">visualize</span> <span m="1280360">what's</span> <span m="1280670">going</span> <span m="1280920">on.</span> <span m="1281320">Suppose</span> <span m="1281920">there</span> <span m="1282150">are</span> <span m="1282350">a</span> <span m="1282390">million</span> <span m="1283050">doors.</span> <span m="1284540">You</span> <span m="1284870">pick</span> <span m="1285410">door</span> <span m="1285640">number</span> <span m="1285880">one.</span> <span m="1286780">Then</span> <span m="1286970">the</span> <span m="1287080">host,</span> <span m="1287660">who</span> <span m="1287760">knows</span> <span m="1288130">what's</span> <span m="1288310">behind</span> <span m="1288650">the</span> <span m="1288720">doors,</span> <span m="1289480">reveals</span> <span m="1290070">every</span> <span m="1290600">other</span> <span m="1290820">door</span> <span m="1292160">but</span> <span m="1292350">one</span> <span m="1292800">and</span> <span m="1293300">777,777.</span> <span m="1296710">Well,</span> <span m="1297020">it would</span> <span m="1297130">sort</span> <span m="1297310">of</span> <span m="1297360">look</span> <span m="1297630">like</span> <span m="1298090">777,777</span> <span m="1299900">is</span> <span m="1300000">the</span> <span m="1300100">one</span> <span m="1300360">with</span> <span m="1300500">the</span> <span m="1300580">car,</span> <span m="1301110">and</span> <span m="1301290">you'd</span> <span m="1301500">switch</span> <span m="1301870">pretty</span> <span m="1302150">fast.</span> </p>
<p><span m="1303205">All</span> <span m="1303660">right</span> <span m="1303800">that's</span> <span m="1304040">her</span> <span m="1304180">argument.</span> <span m="1304620">That</span> <span m="1304790">is</span> <span m="1304920">not</span> <span m="1305180">exactly</span> <span m="1305730">a</span> <span m="1305800">proof,</span> <span m="1307110">OK,</span> <span m="1307410">that</span> <span m="1307550">the</span> <span m="1307700">answer</span> <span m="1307990">is</span> <span m="1308100">you</span> <span m="1308210">should</span> <span m="1308410">switch</span> <span m="1309400">or</span> <span m="1309670">that the</span> <span m="1309980">probability</span> <span m="1310580">is</span> <span m="1310720">2/3.</span> <span m="1312450">All</span> <span m="1312480">right?</span> <span m="1313270">Now,</span> <span m="1313490">because</span> <span m="1313690">she's</span> <span m="1313880">giving</span> <span m="1314200">another</span> <span m="1314740">problem</span> <span m="1315140">with</span> <span m="1315280">a</span> <span m="1315330">million</span> <span m="1315690">doors,</span> <span m="1316170">but</span> <span m="1316350">you've</span> <span m="1316480">got</span> <span m="1316630">a</span> <span m="1316690">problem</span> <span m="1317040">with</span> <span m="1317170">three</span> <span m="1317390">doors.</span> <span m="1318070">And</span> <span m="1318300">as</span> <span m="1318400">we'll</span> <span m="1318520">see</span> <span m="1318700">in</span> <span m="1318800">recitation</span> <span m="1319430">tomorrow,</span> <span m="1320830">the</span> <span m="1320920">number</span> <span m="1321180">of</span> <span m="1321240">doors</span> <span m="1321530">actually</span> <span m="1321960">matters</span> <span m="1322410">in</span> <span m="1322500">terms</span> <span m="1322750">of</span> <span m="1322840">some</span> <span m="1323060">strategies</span> <span m="1323680">here.</span> </p>
<p><span m="1326540">So</span> <span m="1327060">people</span> <span m="1327430">weren't</span> <span m="1327870">convinced</span> <span m="1328410">by</span> <span m="1328610">this.</span> <span m="1329760">And</span> <span m="1330040">so,</span> <span m="1330150">if</span> <span m="1330210">you</span> <span m="1330330">look</span> <span m="1330510">to</span> <span m="1330610">the</span> <span m="1330710">next</span> <span m="1331000">page,</span> <span m="1331510">you</span> <span m="1331590">see</span> <span m="1331880">people</span> <span m="1332200">writing</span> <span m="1332590">in.</span> <span m="1332930">And</span> <span m="1333150">there's</span> <span m="1333370">a</span> <span m="1333420">lot</span> <span m="1333710">a</span> <span m="1333790">write-ins.</span> <span m="1335340">The</span> <span m="1335440">first</span> <span m="1335770">one's</span> <span m="1336040">by</span> <span m="1336690">Robert</span> <span m="1337050">Sachs,</span> <span m="1338070">who's</span> <span m="1338380">actually--</span> <span m="1338830">I</span> <span m="1339200">looked</span> <span m="1339400">him</span> <span m="1339520">up</span> <span m="1339670">on</span> <span m="1339810">Google,</span> <span m="1340700">and</span> <span m="1341130">he's</span> <span m="1341360">on</span> <span m="1341460">the</span> <span m="1341540">faculty</span> <span m="1342030">at</span> <span m="1342110">George</span> <span m="1342450">Mason</span> <span m="1342840">University.</span> <span m="1345450">I</span> <span m="1345620">also</span> <span m="1346270">noticed</span> <span m="1346680">that</span> <span m="1346780">in</span> <span m="1346870">his</span> <span m="1347050">history,</span> <span m="1347390">he</span> <span m="1347500">got</span> <span m="1347660">his</span> <span m="1347800">BA</span> <span m="1348210">from</span> <span m="1348390">Harvard.</span> <span m="1349400">So</span> <span m="1349500">that</span> <span m="1349670">should</span> <span m="1349820">tell</span> <span m="1349980">you</span> <span m="1350100">something</span> <span m="1350710">about</span> <span m="1351390">the</span> <span m="1351480">quality</span> <span m="1352130">of</span> <span m="1352330">his</span> <span m="1352560">response</span> <span m="1353150">here.</span> <span m="1354400">He</span> <span m="1354580">says,</span> <span m="1355420">since</span> <span m="1355700">you</span> <span m="1355870">seem</span> <span m="1356250">to</span> <span m="1356380">enjoy</span> <span m="1356710">coming</span> <span m="1357030">straight</span> <span m="1357400">to</span> <span m="1357480">the</span> <span m="1357570">point,</span> <span m="1358330">I'll</span> <span m="1358530">do</span> <span m="1358630">the</span> <span m="1358760">same.</span> <span m="1359560">You</span> <span m="1359770">blew</span> <span m="1360050">it.</span> <span m="1360890">Let</span> <span m="1361100">me</span> <span m="1361250">explain.</span> <span m="1362170">If</span> <span m="1362250">one</span> <span m="1362530">door</span> <span m="1362830">is shown</span> <span m="1363160">to</span> <span m="1363200">be</span> <span m="1363310">a</span> <span m="1363370">loser,</span> <span m="1364120">that</span> <span m="1364380">information</span> <span m="1365800">changes</span> <span m="1366320">the</span> <span m="1366420">probability</span> <span m="1367120">of</span> <span m="1367300">either</span> <span m="1367620">remaining</span> <span m="1368090">choice,</span> <span m="1369210">neither</span> <span m="1369620">of</span> <span m="1369700">which</span> <span m="1369940">has</span> <span m="1370160">any</span> <span m="1370470">reason</span> <span m="1370850">to</span> <span m="1370920">be</span> <span m="1371040">more</span> <span m="1371330">likely.</span> <span m="1373210">And</span> <span m="1373440">so,</span> <span m="1373570">the</span> <span m="1373750">answer</span> <span m="1374040">is</span> <span m="1374140">one</span> <span m="1374480">half</span> <span m="1374820">of</span> <span m="1374920">having</span> <span m="1375150">the</span> <span m="1375250">car.</span> <span m="1376340">Then</span> <span m="1376510">he</span> <span m="1376600">says,</span> <span m="1377140">as</span> <span m="1377360">a</span> <span m="1377430">professional</span> <span m="1377960">mathematician,</span> <span m="1378830">I'm</span> <span m="1379040">very</span> <span m="1379330">concerned</span> <span m="1379810">with</span> <span m="1379900">the</span> <span m="1379990">general</span> <span m="1380390">public's</span> <span m="1380830">lack</span> <span m="1381210">of</span> <span m="1381310">mathematical</span> <span m="1381880">skills.</span> <span m="1382760">Please</span> <span m="1383080">help</span> <span m="1383370">by</span> <span m="1383480">confessing</span> <span m="1384040">your</span> <span m="1384220">error.</span> <span m="1384545">And</span> <span m="1384870">in</span> <span m="1385030">the</span> <span m="1385130">future,</span> <span m="1385570">be</span> <span m="1385830">more</span> <span m="1386100">careful.</span> <span m="1387090">Just</span> <span m="1387300">a</span> <span m="1387340">little</span> <span m="1387550">gratuitous</span> <span m="1388300">extra</span> <span m="1388860">statement</span> <span m="1389250">there.</span> </p>
<p><span m="1390810">Next</span> <span m="1391050">we've</span> <span m="1391130">got</span> <span m="1391290">a</span> <span m="1391360">PhD</span> <span m="1391940">from</span> <span m="1392120">Florida,</span> <span m="1393110">Scott</span> <span m="1393460">Smith,</span> <span m="1393810">who</span> <span m="1393920">says,</span> <span m="1394270">you</span> <span m="1394460">blew</span> <span m="1394750">it,</span> <span m="1395180">and</span> <span m="1395330">you</span> <span m="1395460">blew</span> <span m="1395680">it</span> <span m="1395760">big.</span> <span m="1396600">Since</span> <span m="1396820">you</span> <span m="1396910">seem</span> <span m="1397230">to</span> <span m="1397290">have</span> <span m="1397590">difficulty</span> <span m="1398210">grasping</span> <span m="1398860">the</span> <span m="1398950">basic</span> <span m="1399390">principles</span> <span m="1399870">at</span> <span m="1399980">work</span> <span m="1400240">here,</span> <span m="1400790">I'll</span> <span m="1401060">explain.</span> <span m="1402350">After</span> <span m="1402650">the</span> <span m="1402750">host</span> <span m="1403080">reveals</span> <span m="1403400">a</span> <span m="1403450">goat,</span> <span m="1404080">you</span> <span m="1404200">now</span> <span m="1404440">have</span> <span m="1404610">a</span> <span m="1404700">one-in-two</span> <span m="1405220">chance</span> <span m="1405560">of</span> <span m="1405650">being</span> <span m="1406020">correct.</span> <span m="1406890">Whether</span> <span m="1407130">you</span> <span m="1407260">change</span> <span m="1407640">your</span> <span m="1407760">selection</span> <span m="1408250">or</span> <span m="1408430">not,</span> <span m="1409530">the</span> <span m="1409630">odds</span> <span m="1409960">are</span> <span m="1410000">the</span> <span m="1410130">same.</span> <span m="1411150">There</span> <span m="1411340">is</span> <span m="1411430">enough</span> <span m="1411810">mathematical</span> <span m="1412460">illiteracy</span> <span m="1413210">in</span> <span m="1413290">this</span> <span m="1413460">country,</span> <span m="1414140">and</span> <span m="1414270">we</span> <span m="1414360">don't</span> <span m="1414600">need</span> <span m="1414760">the</span> <span m="1414850">world's</span> <span m="1415180">highest</span> <span m="1415700">IQ</span> <span m="1416150">propagating</span> <span m="1416780">more.</span> <span m="1417560">Shame.</span> <span m="1419460">Well, now,</span> <span m="1419700">this</span> <span m="1419880">guy</span> <span m="1420030">is</span> <span m="1420140">right</span> <span m="1420350">about</span> <span m="1420590">that</span> <span m="1420770">one</span> <span m="1421000">point</span> <span m="1421240">about</span> <span m="1421510">mathematical</span> <span m="1422140">illiteracy</span> <span m="1422840">in</span> <span m="1422950">the</span> <span m="1423050">country.</span> </p>
<p><span m="1424370">The</span> <span m="1424530">next</span> <span m="1424880">guy,</span> <span m="1425500">Barry</span> <span m="1425780">Pasternack,</span> <span m="1427025">you know,</span> <span m="1427290">he lists</span> <span m="1427610">himself</span> <span m="1427860">as</span> <span m="1428000">California</span> <span m="1428560">Faculty</span> <span m="1428980">Association.</span> <span m="1430360">He</span> <span m="1430500">actually</span> <span m="1430830">was</span> <span m="1430990">president</span> <span m="1431790">of</span> <span m="1431900">the</span> <span m="1432000">California</span> <span m="1432760">Faculty</span> <span m="1433230">Association.</span> <span m="1435050">He</span> <span m="1435300">says,</span> <span m="1435610">your</span> <span m="1435830">answer</span> <span m="1436220">to the</span> <span m="1436340">question</span> <span m="1436800">is</span> <span m="1436950">an</span> <span m="1437050">error,</span> <span m="1437870">but</span> <span m="1438080">if</span> <span m="1438170">it's</span> <span m="1438360">any</span> <span m="1438560">constellation,</span> <span m="1439350">many</span> <span m="1439620">of</span> <span m="1439660">my</span> <span m="1439810">academic</span> <span m="1440370">colleagues</span> <span m="1441120">have</span> <span m="1441280">also</span> <span m="1441570">been</span> <span m="1441740">stumped</span> <span m="1442200">by</span> <span m="1442360">this</span> <span m="1442560">problem.</span> </p>
<p><span m="1444270">OK.</span> <span m="1444580">So</span> <span m="1444710">next</span> <span m="1444990">page,</span> <span m="1445340">Marilyn</span> <span m="1446230">writes</span> <span m="1446550">back.</span> <span m="1447360">And</span> <span m="1447700">she</span> <span m="1447830">says,</span> <span m="1448570">you know,</span> <span m="1448890">good</span> <span m="1449120">heavens,</span> <span m="1449510">with</span> <span m="1449640">so</span> <span m="1449830">much</span> <span m="1450200">learned</span> <span m="1450490">opposition,</span> <span m="1451740">I'll bet</span> <span m="1451950">this</span> <span m="1452110">is</span> <span m="1452190">going</span> <span m="1452320">to</span> <span m="1452370">keep</span> <span m="1452590">math</span> <span m="1452850">classes</span> <span m="1453220">all</span> <span m="1453400">over</span> <span m="1453550">the</span> <span m="1453680">country</span> <span m="1454030">busy</span> <span m="1454350">on</span> <span m="1454480">Monday.</span> <span m="1454960">And</span> <span m="1455120">it</span> <span m="1455230">did.</span> <span m="1455760">It</span> <span m="1455870">was</span> <span m="1456010">a</span> <span m="1456070">huge</span> <span m="1456790">uproar</span> <span m="1457240">at the</span> <span m="1457370">time</span> <span m="1457610">about</span> <span m="1457930">this.</span> <span m="1458910">So</span> <span m="1459030">now</span> <span m="1459310">she</span> <span m="1459490">tries</span> <span m="1460040">to</span> <span m="1460700">explain</span> <span m="1461290">it</span> <span m="1462030">some</span> <span m="1462230">more</span> <span m="1462620">with</span> <span m="1463030">shells</span> <span m="1463890">and</span> <span m="1464570">stuff</span> <span m="1464890">under</span> <span m="1465060">the</span> <span m="1465170">shells</span> <span m="1465700">and</span> <span m="1465860">so</span> <span m="1466090">forth,</span> <span m="1466430">trying</span> <span m="1466810">to</span> <span m="1467000">argue</span> <span m="1467490">it.</span> <span m="1469040">And</span> <span m="1469160">then</span> <span m="1469330">she</span> <span m="1469520">says,</span> <span m="1470090">just</span> <span m="1470310">try</span> <span m="1470570">the</span> <span m="1470690">experiment</span> <span m="1471890">six</span> <span m="1472180">times</span> <span m="1472750">or</span> <span m="1472900">something,</span> <span m="1473420">and</span> <span m="1473630">you'll</span> <span m="1473790">see.</span> <span m="1475116">You know,</span> <span m="1475550">but,</span> <span m="1475690">it's</span> <span m="1475880">not</span> <span m="1476840">really</span> <span m="1477270">convincing.</span> <span m="1478300">It's</span> <span m="1478520">not</span> <span m="1478740">a</span> <span m="1478790">proof.</span> <span m="1479640">She's</span> <span m="1479770">sort</span> <span m="1479980">of</span> <span m="1480200">waving</span> <span m="1480650">hands</span> <span m="1481090">at it to</span> <span m="1481470">give</span> <span m="1481600">you</span> <span m="1481690">some</span> <span m="1481900">idea</span> <span m="1482220">of why</span> <span m="1482340">she</span> <span m="1482700">thinks</span> <span m="1482930">she's</span> <span m="1483080">right.</span> <span m="1483760">But</span> <span m="1483980">when</span> <span m="1484120">I</span> <span m="1484190">look</span> <span m="1484350">at</span> <span m="1484400">the</span> <span m="1484490">hand</span> <span m="1484760">waving</span> <span m="1485050">here</span> <span m="1485310">versus</span> <span m="1485610">the</span> <span m="1485650">hand</span> <span m="1485890">waving</span> <span m="1486220">on</span> <span m="1486370">those</span> <span m="1486630">faculty,</span> <span m="1488080">well,</span> <span m="1488710">it's</span> <span m="1488950">all</span> <span m="1489120">hand</span> <span m="1489370">waving.</span> <span m="1490640">All</span> <span m="1490760">right.</span> </p>
<p><span m="1490960">So</span> <span m="1491040">you</span> <span m="1491080">go</span> <span m="1491220">to</span> <span m="1491300">the</span> <span m="1491390">next</span> <span m="1491660">page,</span> <span m="1492700">and</span> <span m="1492910">now</span> <span m="1493310">the</span> <span m="1493390">letters</span> <span m="1493740">are</span> <span m="1493780">getting</span> <span m="1494000">a</span> <span m="1494050">little</span> <span m="1494250">nastier.</span> <span m="1495910">We</span> <span m="1496080">got</span> <span m="1496290">another</span> <span m="1496620">University</span> <span m="1497170">of</span> <span m="1497250">Florida</span> <span m="1497610">guy.</span> <span m="1498670">He</span> <span m="1498800">says,</span> <span m="1499310">may</span> <span m="1499480">I</span> <span m="1499590">suggest</span> <span m="1500210">you</span> <span m="1500420">obtain</span> <span m="1500930">and</span> <span m="1501050">refer</span> <span m="1501450">to</span> <span m="1501600">a</span> <span m="1501660">standard</span> <span m="1502190">textbook</span> <span m="1502670">on</span> <span m="1502790">probability</span> <span m="1503860">before</span> <span m="1504180">you</span> <span m="1504280">try</span> <span m="1504550">to</span> <span m="1504650">answer</span> <span m="1505110">a</span> <span m="1505150">question</span> <span m="1505560">of</span> <span m="1505620">this</span> <span m="1505800">type</span> <span m="1506110">again.</span> <span m="1507720">Then</span> <span m="1507980">Robert</span> <span m="1508350">Smith,</span> <span m="1508980">Georgia</span> <span m="1509360">State,</span> <span m="1510380">I</span> <span m="1510480">am</span> <span m="1510620">sure</span> <span m="1511000">you</span> <span m="1511140">will</span> <span m="1511270">receive</span> <span m="1511650">many</span> <span m="1511930">letters</span> <span m="1512280">on</span> <span m="1512370">this</span> <span m="1512540">topic</span> <span m="1512930">from</span> <span m="1513070">high</span> <span m="1513320">school</span> <span m="1513820">and</span> <span m="1514070">college</span> <span m="1514500">students.</span> <span m="1515780">Perhaps</span> <span m="1516220">you</span> <span m="1516300">should</span> <span m="1516480">keep</span> <span m="1516680">a</span> <span m="1516750">few</span> <span m="1517020">addresses</span> <span m="1517530">for</span> <span m="1517700">help</span> <span m="1517970">with</span> <span m="1518370">future</span> <span m="1518560">columns.</span> <span m="1520380">These</span> <span m="1520550">are</span> <span m="1520650">great.</span> </p>
<p><span m="1522190">Now</span> <span m="1522310">this</span> <span m="1522540">next</span> <span m="1522780">guy,</span> <span m="1523060">Ray</span> <span m="1523570">Bobo,</span> <span m="1524800">now,</span> <span m="1525540">he's</span> <span m="1525790">a</span> <span m="1525860">real</span> <span m="1526060">guy.</span> <span m="1526570">I looked</span> <span m="1526800">him</span> <span m="1526940">up, too.</span> <span m="1527400">It's</span> <span m="1527520">not</span> <span m="1527740">fake.</span> <span m="1528150">That's</span> <span m="1528320">his</span> <span m="1528470">real</span> <span m="1528670">name.</span> <span m="1529820">And</span> <span m="1530090">I've</span> <span m="1530130">got</span> <span m="1530280">to</span> <span m="1530360">say,</span> <span m="1530640">if</span> <span m="1531250">my</span> <span m="1531370">name</span> <span m="1531560">was</span> <span m="1531720">Bobo,</span> <span m="1532190">I'd</span> <span m="1532280">think</span> <span m="1532540">twice</span> <span m="1532940">before</span> <span m="1533230">writing</span> <span m="1533640">into</span> <span m="1533820"><i>Parade</i></span> <span m="1534600"><i>Magazine</i>,</span> <span m="1535630">you know,</span> <span m="1536590">especially</span> <span m="1537160">if</span> <span m="1537290">I'm</span> <span m="1537610">wrong.</span> <span m="1539070">Anyway,</span> <span m="1539470">here's</span> <span m="1539690">Bobo.</span> <span m="1541080">You</span> <span m="1541250">are</span> <span m="1541470">utterly</span> <span m="1541830">incorrect</span> <span m="1542540">about</span> <span m="1542760">the</span> <span m="1542850">game</span> <span m="1543130">show</span> <span m="1543380">question.</span> <span m="1544260">And</span> <span m="1544420">I</span> <span m="1544480">hope</span> <span m="1544720">this</span> <span m="1544910">controversy</span> <span m="1545570">will</span> <span m="1545710">call</span> <span m="1545970">some</span> <span m="1546350">public</span> <span m="1546700">attention</span> <span m="1547170">to</span> <span m="1547230">the</span> <span m="1547350">serious</span> <span m="1547980">national</span> <span m="1548360">crisis</span> <span m="1548860">in</span> <span m="1548950">mathematical</span> <span m="1549510">education.</span> <span m="1550415">Well,</span> <span m="1550690">that's</span> <span m="1551200">fair.</span> <span m="1552500">If</span> <span m="1552640">you</span> <span m="1552760">could</span> <span m="1552980">admit</span> <span m="1553160">your</span> <span m="1553360">error,</span> <span m="1553690">you</span> <span m="1553810">will have</span> <span m="1553980">contributed</span> <span m="1554450">constructively</span> <span m="1555230">towards</span> <span m="1555610">a</span> <span m="1555680">solution</span> <span m="1556200">of</span> <span m="1556280">a</span> <span m="1556340">deplorable</span> <span m="1556980">situation.</span> <span m="1558320">Now,</span> <span m="1558570">here's</span> <span m="1558790">[INAUDIBLE].</span> <span m="1559090">How</span> <span m="1559300">many</span> <span m="1559820">irate</span> <span m="1560820">mathematicians</span> <span m="1561490">are</span> <span m="1561590">needed</span> <span m="1561940">to</span> <span m="1562010">get</span> <span m="1562260">you to</span> <span m="1562450">change</span> <span m="1562860">your</span> <span m="1562950">mind?</span> <span m="1564380">Well</span> <span m="1564800">that's</span> <span m="1565270">proof</span> <span m="1565530">by</span> <span m="1565700">intimidation,</span> <span m="1566640">right?</span> </p>
<p><span m="1567794">[LAUGHTER]</span> </p>
<p><span m="1568600">You know,</span> <span m="1568880">it's</span> <span m="1569030">not,</span> <span m="1569540">here's</span> <span m="1569960">the</span> <span m="1570120">reason</span> <span m="1570640">why.</span> <span m="1571420">It's</span> <span m="1571650">just,</span> <span m="1571960">so</span> <span m="1572090">many</span> <span m="1572270">irate</span> <span m="1572610">mathematicians</span> <span m="1573260">said</span> <span m="1573490">it,</span> <span m="1573590">they</span> <span m="1573700">must,</span> <span m="1574130">must</span> <span m="1574560">be</span> <span m="1574690">right.</span> </p>
<p><span m="1576510">Now</span> <span m="1576570">the</span> <span m="1576670">next</span> <span m="1576920">guy.</span> <span m="1577550">I am</span> <span m="1577870">in</span> <span m="1577960">shock</span> <span m="1578440">that</span> <span m="1578540">after</span> <span m="1578650">being</span> <span m="1578910">corrected</span> <span m="1579340">by</span> <span m="1579470">at</span> <span m="1579550">least</span> <span m="1579900">three</span> <span m="1580270">mathematicians,</span> <span m="1581110">you</span> <span m="1581280">still</span> <span m="1581610">did</span> <span m="1581670">not</span> <span m="1582040">see</span> <span m="1582220">your</span> <span m="1582360">mistake.</span> <span m="1583300">As</span> <span m="1583480">if</span> <span m="1583720">three</span> <span m="1583990">mathematicians,</span> <span m="1584700">that's</span> <span m="1584880">sort</span> <span m="1585110">of</span> <span m="1585180">the</span> <span m="1585300">criteria</span> <span m="1585880">for</span> <span m="1586020">correctness.</span> <span m="1588340">Oh,</span> <span m="1588460">this</span> <span m="1588660">guy</span> <span m="1588760">is</span> <span m="1588910">bad.</span> <span m="1591190">Maybe</span> <span m="1591680">women</span> <span m="1592200">look</span> <span m="1592440">at</span> <span m="1592560">math</span> <span m="1592900">problems</span> <span m="1593440">differently</span> <span m="1593930">than</span> <span m="1594130">men.</span> <span m="1594970">Ooh.</span> <span m="1596550">Well,</span> <span m="1597080">maybe</span> <span m="1597560">it's</span> <span m="1597700">a</span> <span m="1597760">good</span> <span m="1598030">thing</span> <span m="1598370">in</span> <span m="1598450">this</span> <span m="1598650">case.</span> <span m="1599220">I</span> <span m="1599380">don't</span> <span m="1599540">know.</span> <span m="1601740">Oh,</span> <span m="1601980">then</span> <span m="1602330">we</span> <span m="1602460">have</span> <span m="1602690">Glen</span> <span m="1603340">Calkins,</span> <span m="1603750">you</span> <span m="1603980">are</span> <span m="1604210">the</span> <span m="1604340">goat.</span> <span m="1606690">Finally,</span> <span m="1607190">this</span> <span m="1607280">guy,</span> <span m="1609020">US</span> <span m="1609340">Army</span> <span m="1609600">Research</span> <span m="1609990">Institute,</span> <span m="1610190">a</span> <span m="1610420">little</span> <span m="1610580">scary.</span> <span m="1611550">you</span> <span m="1611780">made</span> <span m="1612020">a</span> <span m="1612070">mistake,</span> <span m="1612810">but</span> <span m="1612970">look</span> <span m="1613180">at</span> <span m="1613240">the</span> <span m="1613330">positive</span> <span m="1613830">side.</span> <span m="1614230">If</span> <span m="1614340">all</span> <span m="1614570">those</span> <span m="1614800">PhD's</span> <span m="1615430">were</span> <span m="1615600">wrong,</span> <span m="1616360">the</span> <span m="1616470">country</span> <span m="1616840">would</span> <span m="1617000">be</span> <span m="1617130">in</span> <span m="1617240">some</span> <span m="1617410">very</span> <span m="1617820">serious</span> <span m="1618350">trouble.</span> </p>
<p><span m="1621536">All right.</span> <span m="1621980">Now</span> <span m="1622570">Marilyn</span> <span m="1623020">writes</span> <span m="1623270">an</span> <span m="1623340">even</span> <span m="1623510">longer</span> <span m="1624000">response</span> <span m="1625620">because</span> <span m="1625730">she's</span> <span m="1625890">getting</span> <span m="1626140">zillions</span> <span m="1626660">of</span> <span m="1626750">letters,</span> <span m="1627160">90%</span> <span m="1628690">voting</span> <span m="1628970">against</span> <span m="1629410">her,</span> <span m="1629710">that</span> <span m="1629860">she's</span> <span m="1630070">wrong.</span> <span m="1631660">She</span> <span m="1631840">begins</span> <span m="1632310">now,</span> <span m="1632520">suggests</span> <span m="1632970">a</span> <span m="1633050">nationwide</span> <span m="1633780">experiment,</span> <span m="1635130">so</span> <span m="1635300">that,</span> <span m="1635570">nationwide,</span> <span m="1636290">you'd</span> <span m="1636380">do</span> <span m="1636550">the</span> <span m="1636680">sample</span> <span m="1637510">a</span> <span m="1637720">million</span> <span m="1638090">times</span> <span m="1638490">and</span> <span m="1638610">get</span> <span m="1638730">the</span> <span m="1638860">answer,</span> <span m="1639750">which</span> <span m="1639960">really</span> <span m="1640240">wouldn't</span> <span m="1640460">prove</span> <span m="1640690">it</span> <span m="1640740">either.</span> <span m="1641500">Then</span> <span m="1641620">we</span> <span m="1641700">go</span> <span m="1641810">to</span> <span m="1641850">the</span> <span m="1641950">last</span> <span m="1642280">page.</span> <span m="1642580">There's</span> <span m="1642680">one</span> <span m="1643010">last</span> <span m="1643400">letter.</span> <span m="1643630">And</span> <span m="1643720">this</span> <span m="1643870">is</span> <span m="1643970">what</span> <span m="1644200">finished</span> <span m="1644660">the</span> <span m="1645230">controversy</span> <span m="1645860">in</span> <span m="1645910">the</span> <span m="1646000">press.</span> </p>
<p><span m="1647420">Comes</span> <span m="1647650">from</span> <span m="1647830">MIT.</span> <span m="1649350">You are</span> <span m="1649510">indeed</span> <span m="1649870">correct.</span> <span m="1650590">My</span> <span m="1650720">colleagues</span> <span m="1651190">at</span> <span m="1651280">work</span> <span m="1651630">had</span> <span m="1651740">a</span> <span m="1651780">ball</span> <span m="1652140">with</span> <span m="1652230">this</span> <span m="1652400">problem,</span> <span m="1653110">and</span> <span m="1653240">I</span> <span m="1653310">dare</span> <span m="1653630">say</span> <span m="1653880">that</span> <span m="1654050">most</span> <span m="1654390">of</span> <span m="1654470">them,</span> <span m="1655050">including</span> <span m="1655510">me</span> <span m="1655710">at</span> <span m="1655830">first,</span> <span m="1656280">thought</span> <span m="1656580">you</span> <span m="1656680">were</span> <span m="1656800">wrong.</span> <span m="1658480">So</span> <span m="1658940">maybe</span> <span m="1659280">that</span> <span m="1659580">$50,000</span> <span m="1660220">a</span> <span m="1660260">year</span> <span m="1660530">somebody's</span> <span m="1660980">paying</span> <span m="1661260">for</span> <span m="1661400">education</span> <span m="1661870">is</span> <span m="1661970">worth</span> <span m="1662190">something,</span> <span m="1662670">because</span> <span m="1663390">MIT</span> <span m="1663850">came</span> <span m="1664100">in</span> <span m="1664200">and</span> <span m="1664680">agreed</span> <span m="1665060">with her.</span> </p>
<p><span m="1667260">OK.</span> <span m="1668490">It</span> <span m="1668650">turns</span> <span m="1668980">out</span> <span m="1669780">that</span> <span m="1669950">Marilyn</span> <span m="1670620">was</span> <span m="1671150">correct</span> <span m="1671830">in</span> <span m="1671980">her</span> <span m="1672270">statement</span> <span m="1672650">that</span> <span m="1672760">you</span> <span m="1672850">should</span> <span m="1673060">switch.</span> <span m="1674180">And</span> <span m="1674300">if</span> <span m="1674390">you</span> <span m="1674490">switch,</span> <span m="1674930">you</span> <span m="1675020">have</span> <span m="1675160">a</span> <span m="1675230">2/3</span> <span m="1675770">chance</span> <span m="1676290">of</span> <span m="1676410">winning.</span> <span m="1677630">So</span> <span m="1677740">if</span> <span m="1677800">you</span> <span m="1677890">don't</span> <span m="1678140">switch,</span> <span m="1678510">you've</span> <span m="1678570">got</span> <span m="1678730">a</span> <span m="1678790">1/3</span> <span m="1679530">chance</span> <span m="1679850">of</span> <span m="1679940">winning,</span> <span m="1681050">provided</span> <span m="1681850">that</span> <span m="1682040">Monty</span> <span m="1682920">is</span> <span m="1683130">guaranteed</span> <span m="1683750">to</span> <span m="1683850">open</span> <span m="1684120">a</span> <span m="1684190">door</span> <span m="1684470">with</span> <span m="1684610">goat.</span> <span m="1685285">You know</span> <span m="1685700">and</span> <span m="1685820">that's</span> <span m="1686340">the</span> <span m="1686460">assumption</span> <span m="1686850">here.</span> <span m="1687780">And</span> <span m="1688060">the</span> <span m="1688140">proof</span> <span m="1688790">is</span> <span m="1689100">simple.</span> <span m="1690720">Although</span> <span m="1691940">Marilyn's</span> <span m="1692410">reasoning</span> <span m="1693030">wasn't</span> <span m="1693460">so</span> <span m="1693740">convincing,</span> <span m="1694270">I</span> <span m="1694320">didn't</span> <span m="1694540">think.</span> <span m="1695590">And</span> <span m="1695720">all</span> <span m="1695890">those</span> <span m="1696120">PhDs</span> <span m="1696920">that</span> <span m="1697030">wrote</span> <span m="1697240">in</span> <span m="1697360">with</span> <span m="1697460">those</span> <span m="1697650">stupid</span> <span m="1697980">letters,</span> <span m="1698500">they're</span> <span m="1698600">probably</span> <span m="1699000">intelligent</span> <span m="1699550">people.</span> <span m="1700220">After</span> <span m="1700480">all,</span> <span m="1700580">they</span> <span m="1700690">got</span> <span m="1701180">advanced</span> <span m="1701570">degrees</span> <span m="1701860">in</span> <span m="1701920">mathematics,</span> <span m="1702480">and</span> <span m="1702770">most</span> <span m="1703050">of</span> <span m="1703160">them were</span> <span m="1703370">faculty</span> <span m="1703870">teaching</span> <span m="1704450">mathematics.</span> <span m="1705270">A</span> <span m="1705370">little</span> <span m="1705570">scary.</span> </p>
<p><span m="1706880">But</span> <span m="1707080">they</span> <span m="1707640">just</span> <span m="1707950">weren't</span> <span m="1708460">following</span> <span m="1709670">the</span> <span m="1709770">basic</span> <span m="1710100">principles.</span> <span m="1710640">They</span> <span m="1710740">were</span> <span m="1710860">following</span> <span m="1711400">their</span> <span m="1711560">intuition.</span> <span m="1712600">They</span> <span m="1712690">didn't</span> <span m="1713060">go</span> <span m="1713310">through</span> <span m="1713420">the</span> <span m="1713510">basic</span> <span m="1713890">steps</span> <span m="1714220">to</span> <span m="1714310">figure</span> <span m="1714640">out</span> <span m="1715150">the</span> <span m="1715220">probability.</span> <span m="1717070">So</span> <span m="1717200">that's</span> <span m="1717380">what</span> <span m="1717460">we're</span> <span m="1717600">going</span> <span m="1717700">to</span> <span m="1717770">do.</span> <span m="1718420">We're</span> <span m="1718440">going</span> <span m="1718530">to</span> <span m="1718590">go</span> <span m="1718820">through</span> <span m="1718910">the</span> <span m="1719010">basic</span> <span m="1719400">steps now</span> <span m="1720210">and</span> <span m="1720340">see</span> <span m="1720460">how</span> <span m="1720570">to</span> <span m="1720660">solve</span> <span m="1721020">it.</span> <span m="1721140">And</span> <span m="1721230">we</span> <span m="1721330">will</span> <span m="1721440">use</span> <span m="1721780">these</span> <span m="1721970">steps</span> <span m="1722330">to</span> <span m="1722420">solve</span> <span m="1722770">pretty</span> <span m="1722930">much</span> <span m="1723210">every</span> <span m="1723520">problem</span> <span m="1723910">in probability.</span> <span m="1725320">OK?</span> </p>
<p><span m="1726840">So</span> <span m="1726950">the</span> <span m="1727090">key,</span> <span m="1728790">the</span> <span m="1728930">first</span> <span m="1729280">step,</span> <span m="1730240">is</span> <span m="1731020">to</span> <span m="1731170">look</span> <span m="1731810">at</span> <span m="1732110">the</span> <span m="1732270">sample</span> <span m="1732860">space</span> <span m="1733910">of</span> <span m="1734080">possible</span> <span m="1734850">outcomes.</span> <span m="1736512">Let me</span> <span m="1736970">define</span> <span m="1737390">sample</span> <span m="1737810">space</span> <span m="1738210">and</span> <span m="1738300">outcome.</span> <span m="1743080">The</span> <span m="1743180">sample</span> <span m="1743620">space</span> <span m="1750410">for</span> <span m="1751380">an</span> <span m="1751520">experiment,</span> <span m="1752440">or a</span> <span m="1752700">probability</span> <span m="1753550">game--</span> <span m="1757360">you</span> <span m="1757500">can</span> <span m="1757640">treat</span> <span m="1757910">this</span> <span m="1758070">Monty</span> <span m="1758370">Hall</span> <span m="1758660">game</span> <span m="1758900">as</span> <span m="1759010">an</span> <span m="1759100">experiment--</span> <span m="1762000">is</span> <span m="1762500">simply</span> <span m="1762920">the</span> <span m="1763030">set</span> <span m="1766370">of</span> <span m="1766950">all</span> <span m="1767330">possible</span> <span m="1767960">outcomes,</span> <span m="1768990">things</span> <span m="1770110">that</span> <span m="1770220">could</span> <span m="1770310">happen.</span> </p>
<p><span m="1773750">In fact,</span> <span m="1773830">let me define</span> <span m="1775170">an</span> <span m="1775320">outcome,</span> <span m="1776660">also</span> <span m="1777090">known</span> <span m="1777390">as</span> <span m="1777510">an</span> <span m="1777590">atomic</span> <span m="1778040">event</span> <span m="1779040">or</span> <span m="1779180">a sample</span> <span m="1779670">point.</span> <span m="1783580">An</span> <span m="1783690">outcome,</span> <span m="1787030">also</span> <span m="1787400">known</span> <span m="1787770">as</span> <span m="1789600">a</span> <span m="1790400">sample</span> <span m="1790790">point--</span> <span m="1793240">we'll</span> <span m="1793400">use</span> <span m="1793570">those</span> <span m="1793760">terms</span> <span m="1794080">interchangeably--</span> <span m="1799470">consists</span> <span m="1799980">of</span> <span m="1804370">all</span> <span m="1804830">the</span> <span m="1804990">information</span> <span m="1805640">about</span> <span m="1805870">the</span> <span m="1805950">experiment,</span> <span m="1821320">including--</span> <span m="1822790">and</span> <span m="1822920">that's</span> <span m="1823610">after</span> <span m="1823885">it's</span> <span m="1824160">been</span> <span m="1824290">performed--</span> <span m="1839470">including</span> <span m="1840820">the</span> <span m="1840940">values</span> <span m="1841540">of</span> <span m="1841640">all</span> <span m="1842110">random</span> <span m="1842530">choices.</span> </p>
<p><span m="1855780">For</span> <span m="1855940">example,</span> <span m="1858250">say</span> <span m="1858420">we</span> <span m="1858550">want</span> <span m="1858780">to</span> <span m="1858830">know</span> <span m="1860350">the</span> <span m="1860610">probability</span> <span m="1861430">of</span> <span m="1861520">winning</span> <span m="1862330">in</span> <span m="1862450">Monty</span> <span m="1862800">Hall</span> <span m="1863910">if</span> <span m="1864080">you</span> <span m="1864190">switch.</span> <span m="1865486">All right,</span> <span m="1865880">let's</span> <span m="1866080">figure</span> <span m="1866280">that</span> <span m="1866500">out.</span> <span m="1867780">So</span> <span m="1868300">in</span> <span m="1868360">that</span> <span m="1868560">case,</span> <span m="1871650">we're</span> <span m="1871720">going</span> <span m="1871840">to</span> <span m="1871880">define</span> <span m="1872280">an</span> <span m="1872400">outcome</span> <span m="1872830">of</span> <span m="1872880">the</span> <span m="1872940">Monty</span> <span m="1873230">Hall</span> <span m="1873460">game</span> <span m="1875264">or a</span> <span m="1875670">sample</span> <span m="1876070">point--</span> <span m="1889040">and</span> <span m="1889410">I'm</span> <span m="1889620">assuming</span> <span m="1889890">in</span> <span m="1890160">this</span> <span m="1890340">case,</span> <span m="1890740">we</span> <span m="1890920">know</span> <span m="1891110">you're</span> <span m="1891230">going</span> <span m="1891320">to</span> <span m="1891390">switch</span> <span m="1897180">and</span> <span m="1897390">the</span> <span m="1897470">contestant</span> <span m="1898490">switches--</span> <span m="1907620">consists</span> <span m="1907890">of</span> <span m="1908090">three things.</span> <span m="1912710">First,</span> <span m="1913950">the</span> <span m="1914050">box</span> <span m="1914420">with</span> <span m="1914570">the</span> <span m="1914630">prize,</span> <span m="1917970">where</span> <span m="1918190">did</span> <span m="1918350">Carroll</span> <span m="1918740">put</span> <span m="1918970">it?</span> <span m="1920780">Second,</span> <span m="1922160">the</span> <span m="1922260">box</span> <span m="1922790">chosen</span> <span m="1923190">by</span> <span m="1923330">the</span> <span m="1923440">contestant.</span> <span m="1930420">And</span> <span m="1930650">third,</span> <span m="1931700">the</span> <span m="1931810">box</span> <span m="1932180">that was</span> <span m="1932480">revealed.</span> <span m="1938160">Once</span> <span m="1938440">you</span> <span m="1938560">know</span> <span m="1938820">those</span> <span m="1939120">three</span> <span m="1939370">things,</span> <span m="1939720">you</span> <span m="1939860">know</span> <span m="1940120">everything</span> <span m="1940900">about</span> <span m="1941140">what</span> <span m="1941280">happened.</span> <span m="1943150">All right?</span> </p>
<p><span m="1944880">For</span> <span m="1945060">example,</span> <span m="1945770">let's</span> <span m="1945960">look</span> <span m="1946090">at</span> <span m="1946170">a</span> <span m="1946400">typical</span> <span m="1946760">sample</span> <span m="1947140">point.</span> <span m="1950330">Say</span> <span m="1950630">I</span> <span m="1950650">took</span> <span m="1951650">sample</span> <span m="1952010">point,</span> <span m="1953030">or</span> <span m="1953200">outcome,</span> <span m="1956460">2,1,3.</span> <span m="1960330">This</span> <span m="1960550">is</span> <span m="1960660">the</span> <span m="1962010">outcome</span> <span m="1962540">where</span> <span m="1967350">the</span> <span m="1967450">prize</span> <span m="1971470">is</span> <span m="1972160">the</span> <span m="1972720">box</span> <span m="1973010">2.</span> <span m="1978290">The</span> <span m="1978380">player</span> <span m="1982520">picks</span> <span m="1982990">first</span> <span m="1985310">box</span> <span m="1986240">1.</span> <span m="1989130">And</span> <span m="1989620">Carol</span> <span m="1991180">reveals</span> <span m="1994320">box</span> <span m="1995650">3.</span> <span m="1999600">OK,</span> <span m="1999900">now</span> <span m="2000100">in</span> <span m="2000280">this</span> <span m="2000550">case,</span> <span m="2001320">in</span> <span m="2001450">this</span> <span m="2001670">outcome,</span> <span m="2002140">did</span> <span m="2002220">the</span> <span m="2002360">contestant</span> <span m="2002940">win?</span> <span m="2008570">Yes,</span> <span m="2008970">because</span> <span m="2009110">we're</span> <span m="2009270">assuming--</span> <span m="2009900">for</span> <span m="2010060">henceforth,</span> <span m="2010700">for most of the rest of</span> <span m="2011210">the day,</span> <span m="2011430">we're</span> <span m="2011550">assuming</span> <span m="2011950">the</span> <span m="2012040">switch</span> <span m="2012380">case,</span> <span m="2013020">because</span> <span m="2013170">we're</span> <span m="2013230">going</span> <span m="2013310">to</span> <span m="2013420">analyze</span> <span m="2013980">that.</span> <span m="2014360">So that's</span> <span m="2014770">fixed,</span> <span m="2015320">they</span> <span m="2015440">switch.</span> <span m="2015860">They</span> <span m="2015950">come</span> <span m="2016190">in</span> <span m="2016320">knowing</span> <span m="2016570">they're</span> <span m="2016670">going</span> <span m="2016770">to</span> <span m="2016820">switch.</span> </p>
<p><span m="2018470">So</span> <span m="2018860">the</span> <span m="2019130">prize</span> <span m="2019700">is</span> <span m="2019820">in</span> <span m="2019980">2,</span> <span m="2020960">all right?</span> <span m="2021180">The prize is here.</span> <span m="2024590">The</span> <span m="2024700">player</span> <span m="2025060">started</span> <span m="2025510">here.</span> <span m="2026560">Carol</span> <span m="2026890">revealed</span> <span m="2027560">3.</span> <span m="2028770">And</span> <span m="2028890">when</span> <span m="2028980">they</span> <span m="2029080">switched,</span> <span m="2030470">they</span> <span m="2030550">got</span> <span m="2030740">the</span> <span m="2030820">prize.</span> <span m="2032370">So</span> <span m="2032610">in</span> <span m="2032680">this</span> <span m="2032920">outcome,</span> <span m="2033370">the</span> <span m="2033460">player</span> <span m="2033760">won.</span> <span m="2036310">OK.</span> <span m="2037950">Now,</span> <span m="2038310">does</span> <span m="2038500">every</span> <span m="2038800">3-tuple</span> <span m="2039400">here</span> <span m="2040730">correspond</span> <span m="2041400">to</span> <span m="2041580">a</span> <span m="2041720">sample</span> <span m="2042220">point,</span> <span m="2042570">to our</span> <span m="2042710">possible</span> <span m="2043200">outcome?</span> <span m="2045770">No.</span> <span m="2046100">What's</span> <span m="2046340">an</span> <span m="2046390">example</span> <span m="2046910">of</span> <span m="2047000">a</span> <span m="2047410">sample</span> <span m="2047830">point</span> <span m="2048070">with</span> <span m="2048199">numbers</span> <span m="2048550">1,</span> <span m="2048760">2,</span> <span m="2048909">and</span> <span m="2049040">3</span> <span m="2049260">that's</span> <span m="2049719">not.</span> <span m="2052554">Yeah?</span> </p>
<p><span m="2056239">Very</span> <span m="2056690">good.</span> <span m="2057199">Yeah,</span> <span m="2057480">so</span> <span m="2057820">for</span> <span m="2057969">example,</span> <span m="2060760">1,</span> <span m="2061280">2,</span> <span m="2061719">1</span> <span m="2065098">is not a sample point,</span> <span m="2067010">is</span> <span m="2067489">not</span> <span m="2067699">in</span> <span m="2067770">the</span> <span m="2067850">set,</span> <span m="2069060">the</span> <span m="2069489">sample</span> <span m="2069800">space,</span> <span m="2071500">because</span> <span m="2072280">we</span> <span m="2072429">can't</span> <span m="2073580">reveal</span> <span m="2074780">the</span> <span m="2074880">box</span> <span m="2075239">with</span> <span m="2075350">the</span> <span m="2075420">prize.</span> <span m="2077310">Similarly,</span> <span m="2078570">2,</span> <span m="2078770">1,</span> <span m="2079150">1</span> <span m="2080880">is</span> <span m="2081040">not</span> <span m="2081300">a</span> <span m="2081350">sample</span> <span m="2081739">point,</span> <span m="2082060">because</span> <span m="2082389">we</span> <span m="2082510">can't</span> <span m="2082989">reveal</span> <span m="2084210">the</span> <span m="2084380">box</span> <span m="2084889">that</span> <span m="2084969">the</span> <span m="2085070">player</span> <span m="2085409">picked.</span> <span m="2086810">Yeah?</span> </p>
<p><span m="2087467">AUDIENCE : I think</span> <span m="2087945">the first</span> <span m="2088423">two pieces</span> <span m="2088901">of</span> <span m="2089379">data actually</span> <span m="2089857">define the third one,</span> <span m="2090813">because</span> <span m="2092725">once,</span> <span m="2093203">or--</span> <span m="2094159">actually,</span> <span m="2094637">no, if they're the same.</span> </p>
<p><span m="2097210">PROFESSOR: Right,</span> <span m="2097770">so</span> <span m="2098570">if</span> <span m="2098825">the</span> <span m="2099080">player</span> <span m="2099490">picked</span> <span m="2099937">the</span> <span m="2100384">box</span> <span m="2100831">with the</span> <span m="2101278">prize,</span> <span m="2102172">which never happened</span> <span m="2102619">in our</span> <span m="2103066">case.</span> <span m="2104110">For</span> <span m="2104190">example,</span> <span m="2104650">1,</span> <span m="2104910">1</span> <span m="2105260">2</span> <span m="2106340">and</span> <span m="2106530">1, 1,</span> <span m="2106910">3</span> <span m="2107310">are</span> <span m="2107360">both</span> <span m="2107580">possible</span> <span m="2112460">because</span> <span m="2112620">you</span> <span m="2112770">could</span> <span m="2113360">put</span> <span m="2113550">the</span> <span m="2113640">prize</span> <span m="2114050">here,</span> <span m="2115090">the</span> <span m="2115190">player</span> <span m="2115460">could</span> <span m="2115650">pick</span> <span m="2115840">it,</span> <span m="2116230">now</span> <span m="2116400">Carol</span> <span m="2116670">has</span> <span m="2116840">a</span> <span m="2116900">choice</span> <span m="2117240">of</span> <span m="2117330">revealing</span> <span m="2117800">2</span> <span m="2117970">or</span> <span m="2118040">3.</span> <span m="2119960">OK?</span> <span m="2122000">So</span> <span m="2122100">these</span> <span m="2122330">are</span> <span m="2122450">OK.</span> </p>
<p><span m="2128810">All</span> <span m="2128950">right,</span> <span m="2129710">so</span> <span m="2129940">let's</span> <span m="2130220">figure</span> <span m="2130570">out</span> <span m="2131690">a</span> <span m="2131760">nice</span> <span m="2132060">way</span> <span m="2132400">of</span> <span m="2132850">characterizing</span> <span m="2134030">all</span> <span m="2134640">outcomes</span> <span m="2135580">in</span> <span m="2135660">the</span> <span m="2135730">sample</span> <span m="2136060">space.</span> <span m="2137400">Now,</span> <span m="2137720">to</span> <span m="2137840">do</span> <span m="2138070">this,</span> <span m="2138870">we're</span> <span m="2138960">going</span> <span m="2139070">to</span> <span m="2139150">set</span> <span m="2139430">up</span> <span m="2139850">a</span> <span m="2139950">tree.</span> <span m="2140920">And</span> <span m="2141040">we're</span> <span m="2141100">going</span> <span m="2141190">to</span> <span m="2141240">use</span> <span m="2141610">a</span> <span m="2142030">thing</span> <span m="2142270">called</span> <span m="2142570">the</span> <span m="2142650">tree</span> <span m="2142980">method</span> <span m="2143810">to</span> <span m="2144070">construct</span> <span m="2144530">the</span> <span m="2144610">sample</span> <span m="2144980">space,</span> <span m="2154750">getting</span> <span m="2155010">all</span> <span m="2155260">the</span> <span m="2155410">outcomes.</span> <span m="2167870">All</span> <span m="2168100">right.</span> <span m="2168360">So</span> <span m="2169170">the</span> <span m="2169280">first</span> <span m="2169570">thing</span> <span m="2169700">we</span> <span m="2169800">do</span> <span m="2170060">is</span> <span m="2170190">figure</span> <span m="2170530">out</span> <span m="2170920">where</span> <span m="2171230">the</span> <span m="2171340">prize</span> <span m="2171840">is.</span> <span m="2177060">And</span> <span m="2177250">we'll</span> <span m="2177400">represent</span> <span m="2177940">that</span> <span m="2178230">as</span> <span m="2178340">sort</span> <span m="2178640">of</span> <span m="2178690">like a decision</span> <span m="2179150">tree.</span> <span m="2179980">There's</span> <span m="2180190">three</span> <span m="2180380">choices.</span> <span m="2182060">It's in</span> <span m="2182260">box</span> <span m="2182740">1,</span> <span m="2183330">box</span> <span m="2183630">2,</span> <span m="2183890">or</span> <span m="2183980">box</span> <span m="2184330">3.</span> <span m="2186320">Then</span> <span m="2186510">the</span> <span m="2186590">next</span> <span m="2186890">thing</span> <span m="2187200">is</span> <span m="2187600">the</span> <span m="2188930">box</span> <span m="2189340">that</span> <span m="2189410">the</span> <span m="2189500">player picked.</span> <span m="2194210">And</span> <span m="2194420">we'll</span> <span m="2194550">continue</span> <span m="2195020">our</span> <span m="2195120">decision</span> <span m="2195580">tree</span> <span m="2195960">here.</span> <span m="2196640">Each</span> <span m="2196880">of</span> <span m="2196970">these</span> <span m="2197270">now</span> <span m="2197580">has</span> <span m="2198560">three</span> <span m="2198790">branches.</span> <span m="2203910">And</span> <span m="2203940">they're</span> <span m="2204070">labeled</span> <span m="2204380">1,</span> <span m="2204690">2,</span> <span m="2204830">or</span> <span m="2205304">3.</span> </p>
<p><span m="2210520">And</span> <span m="2210700">then,</span> <span m="2210970">the</span> <span m="2211500">last</span> <span m="2211960">step</span> <span m="2212810">is</span> <span m="2213970">which</span> <span m="2214240">box is</span> <span m="2214730">revealed.</span> <span m="2220190">Now,</span> <span m="2220550">in</span> <span m="2220660">this</span> <span m="2220910">case,</span> <span m="2221560">if</span> <span m="2221690">the</span> <span m="2222370">prize is</span> <span m="2222950">in</span> <span m="2223090">one</span> <span m="2223620">and</span> <span m="2223730">the</span> <span m="2223830">player</span> <span m="2224930">chose</span> <span m="2225420">one,</span> <span m="2226280">there's</span> <span m="2226490">two</span> <span m="2226670">possibilities.</span> <span m="2228680">We</span> <span m="2228830">could</span> <span m="2228960">reveal</span> <span m="2229610">2</span> <span m="2230100">or reveal</span> <span m="2230510">3.</span> <span m="2232090">So</span> <span m="2232270">we</span> <span m="2232380">get</span> <span m="2232560">1,</span> <span m="2232860">2,</span> <span m="2233170">3</span> <span m="2233530">here</span> <span m="2235090">and</span> <span m="2235270">1,</span> <span m="2235600">2,</span> <span m="2235920">3</span> <span m="2236470">here.</span> <span m="2238830">What</span> <span m="2238990">about</span> <span m="2239270">here?</span> <span m="2240880">Prize is in</span> <span m="2241360">box</span> <span m="2241790">1,</span> <span m="2243380">box</span> <span m="2243690">2</span> <span m="2243880">is</span> <span m="2243970">revealed--</span> <span m="2245666">sorry,</span> <span m="2246490">player</span> <span m="2246840">picked</span> <span m="2247210">box 2.</span> <span m="2247730">Prize is</span> <span m="2248160">in</span> <span m="2248270">1,</span> <span m="2248880">player</span> <span m="2249160">picks</span> <span m="2249400">2,</span> <span m="2250550">how</span> <span m="2250690">many</span> <span m="2250870">choices</span> <span m="2251380">for</span> <span m="2251510">which</span> <span m="2251850">box</span> <span m="2252140">is</span> <span m="2252240">revealed?</span> <span m="2254940">One.</span> <span m="2255660">And</span> <span m="2256270">which</span> <span m="2256470">box</span> <span m="2256760">is</span> <span m="2256850">going</span> <span m="2256960">to</span> <span m="2257000">be</span> <span m="2257140">revealed?</span> <span m="2259390">3.</span> <span m="2261870">All right,</span> <span m="2261960">this is sample point</span> <span m="2262420">1,</span> <span m="2262920">2,</span> <span m="2263420">3.</span> <span m="2264900">And</span> <span m="2265300">here</span> <span m="2265510">we've</span> <span m="2265640">got</span> <span m="2265920">1,</span> <span m="2266350">3,</span> <span m="2268130">2</span> <span m="2269460">is</span> <span m="2269530">the</span> <span m="2269590">only</span> <span m="2269750">choice.</span> <span m="2273150">All</span> <span m="2273350">right.</span> </p>
<p><span m="2274660">And</span> <span m="2275050">let's</span> <span m="2275330">do--</span> <span m="2275860">maybe</span> <span m="2276000">I'll,</span> <span m="2276460">we</span> <span m="2276520">ought to</span> <span m="2276730">fill the whole thing</span> <span m="2276940">out</span> <span m="2277110">here.</span> <span m="2278140">2,</span> <span m="2278570">1,</span> <span m="2280190">3--</span> <span m="2281260">I'm going</span> <span m="2281340">to</span> <span m="2281410">run</span> <span m="2281590">out</span> <span m="2281680">of</span> <span m="2281740">space,</span> <span m="2282005">but--</span> <span m="2286030">2,</span> <span m="2286750">2,</span> <span m="2287470">now</span> <span m="2287730">this</span> <span m="2288060">splits</span> <span m="2290180">into</span> <span m="2290560">1</span> <span m="2290780">and</span> <span m="2290880">3.</span> <span m="2293110">All</span> <span m="2293150">right?</span> <span m="2293790">So</span> <span m="2294160">we</span> <span m="2294280">have</span> <span m="2294550">here</span> <span m="2296310">2,</span> <span m="2297096">2,</span> <span m="2297490">1,</span> <span m="2300280">2,</span> <span m="2300880">2,</span> <span m="2301500">3.</span> <span m="2303120">Now</span> <span m="2303200">let</span> <span m="2303300">me</span> <span m="2304060">do</span> <span m="2304170">the</span> <span m="2304290">last</span> <span m="2304540">one,</span> <span m="2304640">2,</span> <span m="2304830">3,</span> <span m="2305750">1</span> <span m="2306120">here.</span> <span m="2309390">And</span> <span m="2309510">now</span> <span m="2309720">I'll bring</span> <span m="2309990">these</span> <span m="2310230">up</span> <span m="2311340">over</span> <span m="2311610">here.</span> <span m="2313070">I've</span> <span m="2313250">got</span> <span m="2313690">3,</span> <span m="2314120">1,</span> <span m="2315880">2.</span> <span m="2321880">Then</span> <span m="2322030">I've</span> <span m="2322170">got</span> <span m="2322850">3,</span> <span m="2323610">2,</span> <span m="2324470">1.</span> <span m="2328210">And</span> <span m="2328450">finally,</span> <span m="2328860">I've</span> <span m="2328980">got</span> <span m="2329620">3,</span> <span m="2330046">3</span> <span m="2330900">with</span> <span m="2331060">two</span> <span m="2331220">choices,</span> <span m="2333190">1</span> <span m="2333300">and</span> <span m="2333430">2.</span> <span m="2339770">All right?</span> <span m="2340150">That</span> <span m="2340460">is</span> <span m="2340660">now</span> <span m="2341040">the</span> <span m="2341210">construction</span> <span m="2341870">of</span> <span m="2341930">the</span> <span m="2342040">entire</span> <span m="2342500">sample</span> <span m="2342900">space,</span> <span m="2344670">all</span> <span m="2344880">the</span> <span m="2344990">possible</span> <span m="2345480">outcomes.</span> <span m="2347100">How</span> <span m="2347220">many</span> <span m="2347460">sample</span> <span m="2347930">points</span> <span m="2348300">are</span> <span m="2348440">there</span> <span m="2348610">here?</span> <span m="2348850">How</span> <span m="2349010">many</span> <span m="2349200">possible</span> <span m="2349660">outcomes?</span> </p>
<p><span m="2354000">AUDIENCE : 12</span> </p>
<p><span m="2354690">PROFESSOR: 12.</span> <span m="2356190">1,</span> <span m="2356690">2,</span> <span m="2357190">3, 4, 5,</span> <span m="2357690">6,</span> <span m="2358190">7,</span> <span m="2358690">8,</span> <span m="2359190">9, 10, 11,</span> <span m="2359690">12,</span> <span m="2360340">OK.</span> <span m="2362100">Let's</span> <span m="2362410">see</span> <span m="2362610">which</span> <span m="2362880">ones</span> <span m="2363350">correspond</span> <span m="2364000">to</span> <span m="2364060">victory</span> <span m="2364540">for</span> <span m="2364620">the</span> <span m="2364730">player.</span> <span m="2366570">Let's</span> <span m="2366770">label</span> <span m="2367150">each</span> <span m="2367350">one</span> <span m="2367580">as</span> <span m="2367700">a</span> <span m="2367770">win</span> <span m="2368040">for</span> <span m="2368120">the</span> <span m="2368210">player</span> <span m="2368740">or</span> <span m="2368920">a</span> <span m="2368970">loss</span> <span m="2369360">for</span> <span m="2369430">the</span> <span m="2369520">player.</span> <span m="2371150">What's</span> <span m="2371370">this</span> <span m="2371570">one?</span> <span m="2374280">Yeah.</span> <span m="2374650">Prize is</span> <span m="2374910">in 1,</span> <span m="2375580">the</span> <span m="2375660">player</span> <span m="2376000">chose</span> <span m="2376550">1</span> <span m="2377040">where</span> <span m="2377240">the</span> <span m="2377350">prize</span> <span m="2377720">was,</span> <span m="2378090">and</span> <span m="2378280">then</span> <span m="2378550">switched</span> <span m="2379290">away</span> <span m="2379670">from</span> <span m="2379880">it.</span> <span m="2380690">Loss.</span> <span m="2381300">What</span> <span m="2381400">about</span> <span m="2381550">this</span> <span m="2381730">one?</span> <span m="2383650">Loss,</span> <span m="2384160">same</span> <span m="2384420">reason.</span> <span m="2385140">What</span> <span m="2385320">about</span> <span m="2385540">this</span> <span m="2385740">one?</span> <span m="2388180">That's</span> <span m="2388390">a</span> <span m="2388470">win.</span> <span m="2388760">The</span> <span m="2388840">prize</span> <span m="2389220">is</span> <span m="2389360">here,</span> <span m="2389630">the</span> <span m="2389750">player</span> <span m="2390070">went</span> <span m="2390350">there</span> <span m="2390690">and</span> <span m="2390830">switched,</span> <span m="2391560">so</span> <span m="2391710">they</span> <span m="2391850">wound</span> <span m="2392170">up</span> <span m="2392340">there.</span> <span m="2393520">Win.</span> <span m="2395390">And</span> <span m="2395630">these</span> <span m="2395850">are</span> <span m="2396000">all</span> <span m="2396230">wins</span> <span m="2396620">for</span> <span m="2396700">the</span> <span m="2396800">same</span> <span m="2397120">reason.</span> <span m="2398050">Whoop,</span> <span m="2398360">not</span> <span m="2398500">this</span> <span m="2398680">one.</span> <span m="2401280">2,</span> <span m="2401630">1, 1,</span> <span m="2402330">the</span> <span m="2402420">player</span> <span m="2402840">got</span> <span m="2403190">it right</span> <span m="2403460">to</span> <span m="2403540">start</span> <span m="2403900">with,</span> <span m="2404080">and</span> <span m="2404160">then</span> <span m="2404310">switched</span> <span m="2404700">away.</span> <span m="2405170">That's</span> <span m="2405370">a</span> <span m="2405420">loss.</span> <span m="2406830">Loss.</span> <span m="2408470">Win,</span> <span m="2410370">win,</span> <span m="2410845">win.</span> <span m="2412320">And</span> <span m="2412460">here</span> <span m="2412720">the</span> <span m="2412830">player</span> <span m="2413150">got</span> <span m="2413350">it</span> <span m="2413430">right,</span> <span m="2413750">they</span> <span m="2413860">switched</span> <span m="2414210">away,</span> <span m="2414955">it's a lose.</span> </p>
<p><span m="2416350">All</span> <span m="2416440">right,</span> <span m="2416620">so</span> <span m="2416680">we've</span> <span m="2416810">labeled</span> <span m="2417340">every</span> <span m="2417740">outcome</span> <span m="2418680">as</span> <span m="2418890">to</span> <span m="2419010">whether</span> <span m="2419260">the</span> <span m="2419390">player</span> <span m="2420250">who's</span> <span m="2420410">switching</span> <span m="2421370">wins</span> <span m="2421750">or</span> <span m="2421940">loses.</span> <span m="2423420">How</span> <span m="2423610">many</span> <span m="2424350">sample</span> <span m="2424730">points</span> <span m="2425120">lead</span> <span m="2425350">to</span> <span m="2425410">victory?</span> <span m="2427940">6.</span> <span m="2429780">1,</span> <span m="2430240">2,</span> <span m="2430700">3, 4,</span> <span m="2431160">5, 6</span> <span m="2432370">out</span> <span m="2432540">of</span> <span m="2432640">12.</span> <span m="2434590">Uh-oh.</span> <span m="2436420">Was</span> <span m="2436540">Marilyn</span> <span m="2436920">wrong?</span> <span m="2437290">Is</span> <span m="2437390">it</span> <span m="2437500">really</span> <span m="2437790">Bobo</span> <span m="2438170">was</span> <span m="2438330">right?</span> <span m="2438590">It's</span> <span m="2438710">50-50?</span> <span m="2440890">Half</span> <span m="2441210">the</span> <span m="2441320">sample</span> <span m="2441730">points</span> <span m="2442130">lead</span> <span m="2442360">to</span> <span m="2442430">victory.</span> <span m="2444814">Hm,</span> <span m="2445980">what's</span> <span m="2446180">missing</span> <span m="2446510">here?</span> <span m="2446910">Yeah?</span> </p>
<p><span m="2447932">AUDIENCE : The</span> <span m="2448424">first two are</span> <span m="2449408">identical,</span> <span m="2450392">the</span> <span m="2450884">box with the prize</span> <span m="2451376">and the box chosen first.</span> <span m="2452360">Each</span> <span m="2452852">of</span> <span m="2453344">those</span> <span m="2453836">two</span> <span m="2454328">sample</span> <span m="2454820">points coming from</span> <span m="2455312">that</span> <span m="2455804">are half as likely</span> <span m="2456296">as the</span> <span m="2457772">single</span> <span m="2458264">sample points.</span> </p>
<p><span m="2460232">PROFESSOR: That's</span> <span m="2460750">right. What we're</span> <span m="2461130">missing</span> <span m="2461480">here</span> <span m="2462350">is</span> <span m="2462600">the</span> <span m="2462700">likelihood</span> <span m="2464060">of</span> <span m="2464210">an</span> <span m="2464310">outcome,</span> <span m="2465440">the</span> <span m="2465550">probability</span> <span m="2466290">that</span> <span m="2466590">this</span> <span m="2466810">is</span> <span m="2466930">what</span> <span m="2467110">happens.</span> <span m="2468160">We're</span> <span m="2468250">missing</span> <span m="2468560">one</span> <span m="2468800">more</span> <span m="2469030">thing.</span> <span m="2469860">And</span> <span m="2470300">this</span> <span m="2470450">is</span> <span m="2470550">really</span> <span m="2470830">critical</span> <span m="2471270">when</span> <span m="2471380">you're</span> <span m="2471470">making</span> <span m="2471770">your</span> <span m="2471890">sample</span> <span m="2472280">spaces.</span> <span m="2473630">You</span> <span m="2473760">need</span> <span m="2474070">to</span> <span m="2474150">make</span> <span m="2474330">sure</span> <span m="2474850">you</span> <span m="2475180">get</span> <span m="2475360">the</span> <span m="2475440">probabilities</span> <span m="2476140">assigned</span> <span m="2476550">to</span> <span m="2476590">each</span> <span m="2476840">event.</span> <span m="2477180">They're</span> <span m="2477290">not</span> <span m="2477790">necessarily</span> <span m="2479240">equally</span> <span m="2479700">likely.</span> </p>
<p><span m="2481650">In</span> <span m="2481730">fact,</span> <span m="2482780">what</span> <span m="2482830">we</span> <span m="2482940">need</span> <span m="2483160">to</span> <span m="2483230">do</span> <span m="2483500">is</span> <span m="2483590">to</span> <span m="2483680">construct</span> <span m="2484330">the</span> <span m="2484430">probability</span> <span m="2485060">space.</span> <span m="2486400">So</span> <span m="2486520">lets</span> <span m="2486650">define</span> <span m="2487050">that.</span> <span m="2493870">Your</span> <span m="2494300">probability</span> <span m="2494675">space</span> <span m="2500330">consists</span> <span m="2502720">of</span> <span m="2502900">a</span> <span m="2502960">sample</span> <span m="2503450">space--</span> <span m="2504130">which</span> <span m="2504300">we've</span> <span m="2504460">already</span> <span m="2504690">talked</span> <span m="2505010">about.</span> <span m="2506670">That's</span> <span m="2506900">just</span> <span m="2507120">all</span> <span m="2507270">these</span> <span m="2507400">outcomes--</span> <span m="2512050">and</span> <span m="2512770">a</span> <span m="2512940">probability</span> <span m="2513730">function</span> <span m="2522140">that</span> <span m="2522920">maps--</span> <span m="2523215">and</span> <span m="2523510">we</span> <span m="2523620">call</span> <span m="2523870">this</span> <span m="2524060">function,</span> <span m="2524680">usually,</span> <span m="2525100">PR.</span> <span m="2526460">We'll</span> <span m="2526630">have,</span> <span m="2527010">sometimes</span> <span m="2527380">it'll</span> <span m="2527490">be</span> <span m="2527680">f</span> <span m="2528180">or</span> <span m="2528320">something</span> <span m="2528590">like</span> <span m="2528820">that--</span> <span m="2531500">but</span> <span m="2531680">PR</span> <span m="2532470">maps</span> <span m="2532980">the</span> <span m="2533080">sample</span> <span m="2533570">space,</span> <span m="2535360">S,</span> <span m="2536860">to</span> <span m="2536990">the</span> <span m="2537110">real</span> <span m="2537330">numbers</span> <span m="2541325">such</span> <span m="2541820">that</span> <span m="2544300">first,</span> <span m="2545610">for</span> <span m="2546060">every</span> <span m="2546600">outcome</span> <span m="2547870">in</span> <span m="2548000">the</span> <span m="2548090">sample</span> <span m="2548470">space,</span> <span m="2550220">the</span> <span m="2550240">probability</span> <span m="2552670">is</span> <span m="2552960">between</span> <span m="2553310">0</span> <span m="2553690">and</span> <span m="2553810">1</span> <span m="2555440">of</span> <span m="2555550">that</span> <span m="2555740">outcome.</span> <span m="2557590">And</span> <span m="2557880">2,</span> <span m="2558850">if</span> <span m="2559000">you</span> <span m="2559120">sum</span> <span m="2559590">up</span> <span m="2559810">the</span> <span m="2559900">probabilities</span> <span m="2560570">of</span> <span m="2560660">all</span> <span m="2560810">the</span> <span m="2560960">outcomes,</span> <span m="2561400">you</span> <span m="2561520">get</span> <span m="2561690">1.</span> <span m="2562870">So</span> <span m="2563010">the</span> <span m="2563140">sum</span> <span m="2564280">over</span> <span m="2564560">all</span> <span m="2565170">outcomes</span> <span m="2565525">of</span> <span m="2567570">the</span> <span m="2567670">probability</span> <span m="2568330">of</span> <span m="2568370">the</span> <span m="2568510">outcome</span> <span m="2570100">is</span> <span m="2570260">1.</span> <span m="2572650">All</span> <span m="2572750">right?</span> </p>
<p><span m="2573040">So</span> <span m="2573990">the</span> <span m="2574080">sample</span> <span m="2574470">space</span> <span m="2574790">is</span> <span m="2574890">all</span> <span m="2575050">the</span> <span m="2575190">outcomes.</span> <span m="2575750">The</span> <span m="2575850">probability</span> <span m="2576380">space</span> <span m="2576970">adds</span> <span m="2577460">in--</span> <span m="2577850">has</span> <span m="2578050">this</span> <span m="2578210">extra</span> <span m="2578550">thing--</span> <span m="2578850">the</span> <span m="2578950">probability</span> <span m="2579670">of</span> <span m="2579740">each</span> <span m="2580350">outcome.</span> <span m="2581410">And</span> <span m="2581850">really,</span> <span m="2582130">it</span> <span m="2582210">could</span> <span m="2582320">be</span> <span m="2582450">anything,</span> <span m="2582880">as</span> <span m="2583010">long</span> <span m="2583400">is</span> <span m="2584100">the</span> <span m="2584200">probability</span> <span m="2584670">of</span> <span m="2584730">every</span> <span m="2584950">outcome</span> <span m="2585290">is</span> <span m="2585370">between</span> <span m="2585680">0</span> <span m="2586020">and</span> <span m="2586090">1</span> <span m="2586890">and</span> <span m="2587030">they</span> <span m="2587170">all</span> <span m="2587570">add</span> <span m="2587840">up</span> <span m="2588170">to</span> <span m="2588300">1.</span> <span m="2589140">That's</span> <span m="2589430">it.</span> <span m="2592470">All right.</span> <span m="2592690">Now,</span> <span m="2594010">there's</span> <span m="2594210">a</span> <span m="2594270">natural</span> <span m="2594730">interpretation</span> <span m="2595550">of</span> <span m="2595640">what</span> <span m="2596110">the</span> <span m="2596220">probability</span> <span m="2596790">function</span> <span m="2597200">is.</span> <span m="2598090">And</span> <span m="2598250">that</span> <span m="2598470">is</span> <span m="2598800">the</span> <span m="2598900">probability</span> <span m="2599550">of</span> <span m="2599620">a</span> <span m="2599670">sample</span> <span m="2600080">point,</span> <span m="2601090">well,</span> <span m="2601260">should</span> <span m="2601410">be</span> <span m="2601520">the</span> <span m="2601630">probability</span> <span m="2602200">of</span> <span m="2602270">its</span> <span m="2602420">outcome.</span> <span m="2609170">And</span> <span m="2609310">this</span> <span m="2609440">is</span> <span m="2609560">sort</span> <span m="2609820">of</span> <span m="2609910">where</span> <span m="2610090">you</span> <span m="2610180">get</span> <span m="2610340">the</span> <span m="2610430">to</span> <span m="2610800">Marilyn</span> <span m="2610900">saying,</span> <span m="2611300">hey,</span> <span m="2611470">try</span> <span m="2611840">it</span> <span m="2611910">a</span> <span m="2611970">million</span> <span m="2612290">times</span> <span m="2612700">and you'll</span> <span m="2612890">see</span> <span m="2613080">2/3</span> <span m="2613530">of</span> <span m="2613580">the</span> <span m="2613680">time,</span> <span m="2614000">roughly,</span> <span m="2614380">you</span> <span m="2614450">get</span> <span m="2614600">the</span> <span m="2614690">right,</span> <span m="2615736">that</span> <span m="2616090">you</span> <span m="2616210">win.</span> </p>
<p><span m="2618600">So</span> <span m="2618780">for</span> <span m="2618930">every</span> <span m="2619220">outcome,</span> <span m="2621554">the</span> <span m="2621990">probability</span> <span m="2622990">of</span> <span m="2623080">that</span> <span m="2623270">outcome,</span> <span m="2623730">that</span> <span m="2624330">real</span> <span m="2624570">number,</span> <span m="2626020">should</span> <span m="2626270">be</span> <span m="2626600">the</span> <span m="2626720">probability</span> <span m="2632270">that</span> <span m="2632450">when</span> <span m="2632540">you</span> <span m="2632670">run</span> <span m="2632940">the</span> <span m="2633050">experiment,</span> <span m="2634000">w</span> <span m="2634520">is</span> <span m="2634620">what</span> <span m="2634760">happens.</span> <span m="2639270">w</span> <span m="2639650">will</span> <span m="2639960">be</span> <span m="2640210">the</span> <span m="2640420">outcome.</span> <span m="2647090">In</span> <span m="2647160">other</span> <span m="2647300">words,</span> <span m="2647630">if</span> <span m="2647690">probability</span> <span m="2648260">of</span> <span m="2648330">w</span> <span m="2648690">were a</span> <span m="2649030">half,</span> <span m="2650580">what</span> <span m="2650730">that</span> <span m="2650910">means</span> <span m="2651400">is</span> <span m="2652500">you</span> <span m="2652620">should</span> <span m="2652900">get</span> <span m="2653160">that</span> <span m="2653490">half</span> <span m="2653860">the</span> <span m="2653960">time,</span> <span m="2654830">roughly</span> <span m="2655220">speaking.</span> <span m="2655670">We'll</span> <span m="2655830">get</span> <span m="2655980">into</span> <span m="2656220">really</span> <span m="2656560">details</span> <span m="2656960">of</span> <span m="2657020">that</span> <span m="2657210">later.</span> <span m="2657915">In other words,</span> <span m="2658230">you</span> <span m="2658440">toss</span> <span m="2658810">a</span> <span m="2658870">fair</span> <span m="2659160">coin,</span> <span m="2659590">you</span> <span m="2659710">flip</span> <span m="2659960">it.</span> <span m="2660810">Well, you</span> <span m="2660990">might</span> <span m="2661180">say, a</span> <span m="2661360">probability</span> <span m="2661830">half</span> <span m="2662170">of</span> <span m="2662300">a</span> <span m="2662330">heads.</span> <span m="2663320">All right?</span> <span m="2663760">And</span> <span m="2663910">so,</span> <span m="2664280">this</span> <span m="2664500">function,</span> <span m="2665030">then,</span> <span m="2665180">the</span> <span m="2665340">probability</span> <span m="2665820">of</span> <span m="2665890">heads</span> <span m="2666640">is</span> <span m="2666790">1/2.</span> </p>
<p><span m="2668620">Now,</span> <span m="2669370">assigning</span> <span m="2670000">the</span> <span m="2670150">right</span> <span m="2670510">probabilities</span> <span m="2671120">to</span> <span m="2671210">the</span> <span m="2671330">outcomes</span> <span m="2672540">is</span> <span m="2673460">95%</span> <span m="2674330">of</span> <span m="2674390">the</span> <span m="2674470">work.</span> <span m="2676410">In</span> <span m="2676540">fact, if you</span> <span m="2676770">screw</span> <span m="2677040">that</span> <span m="2677260">up,</span> <span m="2677460">you're</span> <span m="2677570">doomed</span> <span m="2677910">because</span> <span m="2678440">you</span> <span m="2678580">can't</span> <span m="2678810">get</span> <span m="2678940">the</span> <span m="2679020">right</span> <span m="2679230">answer.</span> <span m="2680660">So</span> <span m="2680800">for</span> <span m="2680910">example,</span> <span m="2681250">if</span> <span m="2681320">we</span> <span m="2681430">just</span> <span m="2681580">said,</span> <span m="2681850">OK,</span> <span m="2682120">each</span> <span m="2682350">of</span> <span m="2682420">these</span> <span m="2682600">is</span> <span m="2682720">probability</span> <span m="2683180">1/12,</span> <span m="2685170">we'd</span> <span m="2685350">get</span> <span m="2685530">Bobo's</span> <span m="2685980">answer</span> <span m="2686720">as</span> <span m="2686790">the</span> <span m="2686880">probability</span> <span m="2687360">of</span> <span m="2687440">winning.</span> <span m="2687820">So</span> <span m="2687960">that's</span> <span m="2688210">not</span> <span m="2688480">too</span> <span m="2688600">good.</span> </p>
<p><span m="2691010">Now,</span> <span m="2691190">to</span> <span m="2691330">get</span> <span m="2691650">the</span> <span m="2691760">right</span> <span m="2692030">probabilities</span> <span m="2692540">for</span> <span m="2692670">the</span> <span m="2692750">Monty</span> <span m="2693050">Hall</span> <span m="2693320">problem,</span> <span m="2694600">we</span> <span m="2694740">have</span> <span m="2695050">to</span> <span m="2695160">make</span> <span m="2695480">some</span> <span m="2695620">assumptions.</span> <span m="2698690">So</span> <span m="2698750">let's</span> <span m="2699050">write</span> <span m="2699260">those</span> <span m="2699757">down.</span> <span m="2724140">So</span> <span m="2724420">the</span> <span m="2724540">first</span> <span m="2724870">assumption</span> <span m="2725310">we're</span> <span m="2725440">going</span> <span m="2725550">to</span> <span m="2725610">make</span> <span m="2726710">is</span> <span m="2727170">that</span> <span m="2727780">Carol</span> <span m="2728290">put</span> <span m="2728490">the</span> <span m="2728580">prize</span> <span m="2729100">in</span> <span m="2729190">a</span> <span m="2729270">random</span> <span m="2729640">box.</span> <span m="2730170">They're</span> <span m="2730320">all</span> <span m="2730850">equally</span> <span m="2731210">likely.</span> <span m="2733740">So</span> <span m="2733840">the</span> <span m="2733950">prize</span> <span m="2736570">is</span> <span m="2736880">in</span> <span m="2738700">each</span> <span m="2739000">box</span> <span m="2741930">with</span> <span m="2742180">probability</span> <span m="2744100">1/3.</span> <span m="2747040">The</span> <span m="2747150">next</span> <span m="2747480">assumption</span> <span m="2747920">we're</span> <span m="2748040">going</span> <span m="2748130">to</span> <span m="2748200">make</span> <span m="2749380">is</span> <span m="2749640">that</span> <span m="2749720">the</span> <span m="2749810">player</span> <span m="2750260">didn't</span> <span m="2750550">see</span> <span m="2751920">which</span> <span m="2752160">box</span> <span m="2752460">has</span> <span m="2752670">the</span> <span m="2752730">prize.</span> <span m="2754440">OK?</span> <span m="2754700">In</span> <span m="2754790">other</span> <span m="2754960">words,</span> <span m="2755340">and</span> <span m="2755600">more</span> <span m="2755790">specifically,</span> <span m="2758110">no</span> <span m="2758230">matter</span> <span m="2758550">where</span> <span m="2758780">the</span> <span m="2758890">prize</span> <span m="2759190">is,</span> <span m="2767280">the</span> <span m="2767400">player</span> <span m="2768350">picks</span> <span m="2768810">each</span> <span m="2769120">box</span> <span m="2769520">with</span> <span m="2769630">probability of</span> <span m="2770110">1/3.</span> <span m="2771380">So</span> <span m="2771400">the</span> <span m="2771490">player's</span> <span m="2771900">making,</span> <span m="2772650">effectively,</span> <span m="2773070">a</span> <span m="2773150">random</span> <span m="2773440">choice.</span> <span m="2787449">All right,</span> <span m="2787946">and</span> <span m="2788450">there's</span> <span m="2788590">one</span> <span m="2788850">more</span> <span m="2789090">assumption</span> <span m="2789550">we're</span> <span m="2789670">going</span> <span m="2789770">to</span> <span m="2789830">make.</span> <span m="2790610">Anybody</span> <span m="2790900">have</span> <span m="2791060">any</span> <span m="2791180">ideas</span> <span m="2791520">about</span> <span m="2791720">what's</span> <span m="2791940">left</span> <span m="2792190">to</span> <span m="2792270">make</span> <span m="2792480">as an</span> <span m="2792620">assumption</span> <span m="2793090">here?</span> <span m="2794812">Yeah?</span> </p>
<p><span m="2798130">AUDIENCE : I guess,</span> <span m="2798570">if</span> <span m="2798830">Carol's</span> <span m="2799030">the</span> <span m="2799210">one</span> <span m="2799690">who's</span> <span m="2799750">opening</span> <span m="2799890">the</span> <span m="2800170">boxes,</span> <span m="2800290">she knows</span> <span m="2800520">where all the prizes</span> <span m="2800995">are.</span> </p>
<p><span m="2802290">PROFESSOR: Yeah.</span> <span m="2803110">OK.</span> <span m="2803740">So</span> <span m="2804240">what</span> <span m="2804740">assumption</span> <span m="2805240">should</span> <span m="2805740">I make?</span> <span m="2810980">Yeah, that she</span> <span m="2811060">picks a box</span> <span m="2812080">at</span> <span m="2812330">random</span> <span m="2813200">if</span> <span m="2813350">she</span> <span m="2813470">has</span> <span m="2813670">a</span> <span m="2813720">choice.</span> <span m="2815251">All</span> <span m="2815670">right,</span> <span m="2816050">so</span> <span m="2816120">we're</span> <span m="2816180">going</span> <span m="2816260">to</span> <span m="2816300">make</span> <span m="2816490">that</span> <span m="2816650">assumption.</span> <span m="2817160">So</span> <span m="2817360">that</span> <span m="2818120">no</span> <span m="2818270">matter</span> <span m="2823200">where</span> <span m="2823650">the</span> <span m="2823750">prize</span> <span m="2824220">is,</span> <span m="2829180">if</span> <span m="2829420">Carol</span> <span m="2829850">has</span> <span m="2830060">a</span> <span m="2830120">choice,</span> <span m="2837230">she</span> <span m="2837450">picks</span> <span m="2839740">each</span> <span m="2840040">box</span> <span m="2840520">with</span> <span m="2840670">probability</span> <span m="2841120">1/2.</span> </p>
<p><span m="2855960">Now,</span> <span m="2856250">why</span> <span m="2856510">is</span> <span m="2856710">this</span> <span m="2856960">third</span> <span m="2857250">one</span> <span m="2857470">important?</span> <span m="2858430">Why</span> <span m="2858650">might</span> <span m="2858900">that</span> <span m="2859170">really</span> <span m="2859420">matter?</span> <span m="2861490">Can anybody</span> <span m="2861650">think of a</span> <span m="2861810">reason</span> <span m="2862310">why that</span> <span m="2862810">could be</span> <span m="2863310">relevant?</span> <span m="2866358">What</span> <span m="2866836">if</span> <span m="2867314">Carol</span> <span m="2867792">had</span> <span m="2868270">some</span> <span m="2869226">strategy</span> <span m="2869704">in mind</span> <span m="2870182">that the</span> <span m="2870660">contestant</span> <span m="2871138">knew about?</span> <span m="2872572">Can you</span> <span m="2873050">think of</span> <span m="2873210">why that might</span> <span m="2873650">be useful.</span> </p>
<p><span m="2874896">AUDIENCE : If the</span> <span m="2875862">contestant</span> <span m="2876345">were able to</span> <span m="2877311">produce</span> <span m="2877794">information</span> <span m="2878277">based off of</span> <span m="2878760">that,</span> <span m="2879726">then there's a</span> <span m="2880209">higher</span> <span m="2880692">chance a contestant</span> <span m="2881175">could figure</span> <span m="2881658">out which one</span> <span m="2882141">to choose.</span> </p>
<p><span m="2883107">PROFESSOR: Yeah.</span> <span m="2884073">In fact,</span> <span m="2886510">several</span> <span m="2887280">years</span> <span m="2887700">ago,</span> <span m="2888120">when</span> <span m="2888280">I</span> <span m="2888330">was</span> <span m="2888510">lecturing,</span> <span m="2889010">I</span> <span m="2889060">said,</span> <span m="2889330">in</span> <span m="2889410">fact</span> <span m="2889650">it</span> <span m="2889710">could</span> <span m="2889850">make</span> <span m="2890020">a</span> <span m="2890050">difference.</span> <span m="2891770">That</span> <span m="2891910">there</span> <span m="2892090">is</span> <span m="2892290">a</span> <span m="2892370">way</span> <span m="2892600">that</span> <span m="2892750">Carol</span> <span m="2893150">can</span> <span m="2893340">convey</span> <span m="2893670">information</span> <span m="2894220">to</span> <span m="2894270">the</span> <span m="2894380">player</span> <span m="2895410">by</span> <span m="2895800">of</span> <span m="2895930">two</span> <span m="2896150">she's</span> <span m="2896390">choosing,</span> <span m="2896840">which</span> <span m="2897020">one</span> <span m="2897210">to</span> <span m="2897280">pick.</span> <span m="2897930">Like,  if</span> <span m="2898260">she has</span> <span m="2898410">a</span> <span m="2898470">choice</span> <span m="2898740">of</span> <span m="2898840">2</span> <span m="2899030">or</span> <span m="2899100">3,</span> <span m="2900150">say</span> <span m="2900340">she</span> <span m="2900530">always</span> <span m="2900890">picks</span> <span m="2901270">bigger</span> <span m="2901560">one,</span> <span m="2901750">3.</span> <span m="2902790">Well, that</span> <span m="2903010">might</span> <span m="2903190">tell</span> <span m="2903320">you</span> <span m="2903820">information</span> <span m="2904430">about,</span> <span m="2904920">she</span> <span m="2905350">didn't</span> <span m="2905560">pick</span> <span m="2905750">2,</span> <span m="2905970">so</span> <span m="2906120">maybe</span> <span m="2906350">it</span> <span m="2906430">has</span> <span m="2906660">to</span> <span m="2906730">have,</span> <span m="2907370">or she--</span> <span m="2908010">well,</span> <span m="2908580">I don't</span> <span m="2908600">know.</span> </p>
<p><span m="2910460">In fact,</span> <span m="2910600">that's</span> <span m="2910870">wrong,</span> <span m="2912110">and</span> <span m="2912550">it</span> <span m="2912650">doesn't</span> <span m="2912970">matter</span> <span m="2913900">in</span> <span m="2914010">the</span> <span m="2914090">case</span> <span m="2914360">of</span> <span m="2914470">2</span> <span m="2914690">boxes,</span> <span m="2915360">this</span> <span m="2915540">assumption.</span> <span m="2916030">We're</span> <span m="2916200">going</span> <span m="2916290">to</span> <span m="2916360">make</span> <span m="2916600">it,</span> <span m="2916790">because</span> <span m="2916840">we're</span> <span m="2916940">going</span> <span m="2917030">to</span> <span m="2917090">do</span> <span m="2917200">the</span> <span m="2917320">analysis</span> <span m="2917790">with</span> <span m="2917960">it.</span> <span m="2918900">But</span> <span m="2919070">for</span> <span m="2919160">three</span> <span m="2919400">boxes,</span> <span m="2920150">you</span> <span m="2920300">can</span> <span m="2920450">show</span> <span m="2920750">that</span> <span m="2920850">it</span> <span m="2920960">really</span> <span m="2921220">doesn't</span> <span m="2921640">matter</span> <span m="2922410">for</span> <span m="2922530">this</span> <span m="2922710">assumption.</span> <span m="2923650">But</span> <span m="2923700">if</span> <span m="2923760">you</span> <span m="2923840">go</span> <span m="2923960">to</span> <span m="2924050">four</span> <span m="2924310">boxes,</span> <span m="2925200">it</span> <span m="2925350">does.</span> <span m="2925820">And</span> <span m="2926300">so</span> <span m="2926440">in</span> <span m="2926500">recitation</span> <span m="2927110">tomorrow,</span> <span m="2927400">you're</span> <span m="2927510">going</span> <span m="2927600">to</span> <span m="2927640">consider</span> <span m="2927970">the</span> <span m="2928070">four</span> <span m="2928370">box</span> <span m="2928710">version</span> <span m="2929000">of</span> <span m="2929060">this</span> <span m="2929230">game.</span> <span m="2930250">And if</span> <span m="2930500">Carol</span> <span m="2930950">has</span> <span m="2931210">a</span> <span m="2931280">strategy</span> <span m="2932080">that</span> <span m="2932220">the</span> <span m="2932310">player</span> <span m="2932670">knows</span> <span m="2932940">about,</span> <span m="2933810">you</span> <span m="2933970">could</span> <span m="2934100">improve</span> <span m="2934520">the</span> <span m="2934600">player's</span> <span m="2935030">chances</span> <span m="2935430">of</span> <span m="2935500">winning.</span> <span m="2937480">But</span> <span m="2937640">not</span> <span m="2938150">for</span> <span m="2938270">three,</span> <span m="2938520">it</span> <span m="2938630">turns</span> <span m="2938900">out.</span> </p>
<p><span m="2939730">In</span> <span m="2939860">any</span> <span m="2940040">case,</span> <span m="2941080">when</span> <span m="2941200">you're</span> <span m="2941770">setting</span> <span m="2942030">up</span> <span m="2942120">a</span> <span m="2942200">problem,</span> <span m="2942690">you've</span> <span m="2942840">got</span> <span m="2943080">to</span> <span m="2943270">set</span> <span m="2943540">up</span> <span m="2944170">your</span> <span m="2944360">assumptions.</span> <span m="2944940">Otherwise,</span> <span m="2945350">you</span> <span m="2945450">don't</span> <span m="2945720">get</span> <span m="2945900">anywhere.</span> <span m="2947530">These</span> <span m="2947740">are</span> <span m="2947800">sort</span> <span m="2947990">of</span> <span m="2948030">like</span> <span m="2948410">axioms,</span> <span m="2950110">same</span> <span m="2950380">kind</span> <span m="2950570">of</span> <span m="2950660">thing.</span> <span m="2950970">It's</span> <span m="2951020">fine</span> <span m="2951370">to</span> <span m="2951430">make</span> <span m="2951630">assumptions.</span> <span m="2952150">Just</span> <span m="2952360">state</span> <span m="2953300">what</span> <span m="2953490">they</span> <span m="2953650">are.</span> <span m="2955750">Now,</span> <span m="2955870">of</span> <span m="2955940">course,</span> <span m="2956310">you</span> <span m="2956410">also</span> <span m="2956650">want</span> <span m="2956880">them to</span> <span m="2956950">be</span> <span m="2957120">reasonable.</span> <span m="2963450">All</span> <span m="2963570">right.</span> <span m="2963800">So</span> <span m="2964180">maybe we're</span> <span m="2964270">going</span> <span m="2964360">to</span> <span m="2964400">go</span> <span m="2964540">back</span> <span m="2964880">now</span> <span m="2965310">and</span> <span m="2965750">figure</span> <span m="2966170">out</span> <span m="2967530">how</span> <span m="2967790">to</span> <span m="2967870">get</span> <span m="2968050">the</span> <span m="2968130">probabilities</span> <span m="2969070">of</span> <span m="2969370">the</span> <span m="2970630">sample</span> <span m="2971100">points,</span> <span m="2971470">or</span> <span m="2971500">the</span> <span m="2971640">outcomes,</span> <span m="2972110">for</span> <span m="2972200">Monty</span> <span m="2972510">Hall.</span> <span m="2974690">Let's</span> <span m="2975070">go do</span> <span m="2975210">that.</span> <span m="2977050">And</span> <span m="2977200">maybe</span> <span m="2977370">I'll</span> <span m="2977500">draw</span> <span m="2977750">them in</span> <span m="2977910">a different</span> <span m="2978398">color here.</span> </p>
<p><span m="2983280">What</span> <span m="2983580">I'm</span> <span m="2983700">going</span> <span m="2983810">to</span> <span m="2983900">do</span> <span m="2984210">is</span> <span m="2984350">label</span> <span m="2984840">every</span> <span m="2985270">edge</span> <span m="2985790">with</span> <span m="2985970">a</span> <span m="2986020">probability</span> <span m="2987220">associated</span> <span m="2987990">with</span> <span m="2988120">taking</span> <span m="2988500">that</span> <span m="2988740">step.</span> <span m="2990990">So</span> <span m="2991280">if</span> <span m="2991400">the</span> <span m="2992920">prize</span> <span m="2993450">is</span> <span m="2993580">placed</span> <span m="2993940">in</span> <span m="2994020">each</span> <span m="2994300">box</span> <span m="2994680">with a</span> <span m="2994760">probability</span> <span m="2995260">of</span> <span m="2995780">1/3,</span> <span m="2997020">that</span> <span m="2997240">says</span> <span m="2997430">this</span> <span m="2997680">happens</span> <span m="2998070">with</span> <span m="2998170">probability</span> <span m="2998610">1/3,</span> <span m="3000100">that's</span> <span m="3000400">1/3,</span> <span m="3001730">and</span> <span m="3001850">that's</span> <span m="3002130">1/3.</span> <span m="3005000">Same</span> <span m="3005320">thing</span> <span m="3005540">here.</span> <span m="3006570">Assumption</span> <span m="3007940">two</span> <span m="3008250">says</span> <span m="3008630">that</span> <span m="3008780">no</span> <span m="3008860">matter</span> <span m="3009110">where</span> <span m="3009410">the</span> <span m="3009540">prize</span> <span m="3010080">is,</span> <span m="3010510">say</span> <span m="3010640">it's in</span> <span m="3010870">box</span> <span m="3011220">1,</span> <span m="3013190">the</span> <span m="3013300">player</span> <span m="3013790">chooses</span> <span m="3014240">each</span> <span m="3014470">box</span> <span m="3014820">with a</span> <span m="3014900">probably</span> <span m="3015330">of</span> <span m="3015640">1/3.</span> <span m="3016680">So</span> <span m="3017230">given</span> <span m="3017550">that the</span> <span m="3017650">prize is</span> <span m="3018280">in</span> <span m="3018370">box</span> <span m="3018710">1,</span> <span m="3019110">the</span> <span m="3019510">player</span> <span m="3019830">chooses</span> <span m="3020270">1</span> <span m="3020540">with</span> <span m="3020690">1/3,</span> <span m="3021900">2 with</span> <span m="3021990">1/3,</span> <span m="3022360">3 with 1/3.</span> <span m="3024330">Same if the</span> <span m="3024710">prize</span> <span m="3025015">is in</span> <span m="3025410">box</span> <span m="3025760">2.</span> <span m="3027410">So</span> <span m="3027520">I</span> <span m="3027620">label</span> <span m="3027950">every</span> <span m="3028240">edge</span> <span m="3028480">here</span> <span m="3028650">with the</span> <span m="3028770">probability.</span> <span m="3031780">All</span> <span m="3031940">right.</span> </p>
<p><span m="3034800">Now</span> <span m="3034980">I</span> <span m="3035060">go</span> <span m="3035210">to</span> <span m="3035290">this</span> <span m="3035690">level.</span> <span m="3036850">And</span> <span m="3037020">I'm,</span> <span m="3037510">at</span> <span m="3037670">this</span> <span m="3037890">node,</span> <span m="3038230">I'm</span> <span m="3038360">in</span> <span m="3038430">the</span> <span m="3038520">state</span> <span m="3039050">where</span> <span m="3040150">prize is in</span> <span m="3040740">box</span> <span m="3041190">1,</span> <span m="3041930">player</span> <span m="3042280">selected</span> <span m="3042830">box</span> <span m="3043170">1,</span> <span m="3043960">so</span> <span m="3044160">Carol</span> <span m="3044520">has</span> <span m="3044720">a</span> <span m="3044790">choice</span> <span m="3045210">of</span> <span m="3045310">2</span> <span m="3045540">or</span> <span m="3045620">3</span> <span m="3046040">to</span> <span m="3046160">reveal.</span> <span m="3048420">And</span> <span m="3048690">she's</span> <span m="3048880">going</span> <span m="3049000">to</span> <span m="3049070">reveal</span> <span m="3049630">each</span> <span m="3049890">one</span> <span m="3050070">with</span> <span m="3050210">probability</span> <span m="3050730">1/2.</span> <span m="3053110">So</span> <span m="3053230">I</span> <span m="3053310">label</span> <span m="3053660">1/2</span> <span m="3053990">on</span> <span m="3054100">each</span> <span m="3054270">of</span> <span m="3054330">these,</span> <span m="3054850">because</span> <span m="3055050">that's</span> <span m="3055250">what</span> <span m="3055330">we've</span> <span m="3055480">assumed.</span> <span m="3057980">What</span> <span m="3058200">probability</span> <span m="3058740">do</span> <span m="3058840">I</span> <span m="3058920">put</span> <span m="3059110">on</span> <span m="3059220">this</span> <span m="3059440">edge</span> <span m="3059900">here?</span> <span m="3063280">Probability</span> <span m="3063730">1,</span> <span m="3064700">because</span> <span m="3064870">there's</span> <span m="3065020">no</span> <span m="3065210">choice.</span> <span m="3065570">Carol</span> <span m="3065850">has</span> <span m="3066090">to</span> <span m="3066180">do</span> <span m="3066360">it.</span> <span m="3067020">The</span> <span m="3067060">same</span> <span m="3067360">here.</span> <span m="3068430">And</span> <span m="3068640">I</span> <span m="3068690">get</span> <span m="3068830">1/2,</span> <span m="3069880">1/2,</span> <span m="3071350">1,</span> <span m="3073610">1,</span> <span m="3074160">1,</span> <span m="3074780">1/2,</span> <span m="3076248">1/2.</span> <span m="3077520">All right,</span> <span m="3077790">so I've</span> <span m="3078030">labeled</span> <span m="3078450">every</span> <span m="3078840">edge</span> <span m="3079040">with</span> <span m="3079140">its</span> <span m="3079290">probability.</span> <span m="3081670">Pretty</span> <span m="3081890">simple,</span> <span m="3082170">because</span> <span m="3082340">it</span> <span m="3082420">was</span> <span m="3082560">all</span> <span m="3082890">assumed.</span> </p>
<p><span m="3084380">Now</span> <span m="3084640">I've</span> <span m="3084790">got</span> <span m="3084930">to</span> <span m="3084970">get</span> <span m="3085130">a</span> <span m="3085190">probability</span> <span m="3085810">for</span> <span m="3085990">each</span> <span m="3086180">outcome.</span> <span m="3087230">And</span> <span m="3087350">we</span> <span m="3087460">do</span> <span m="3087650">that</span> <span m="3087900">with</span> <span m="3088000">the</span> <span m="3088080">following</span> <span m="3088660">rule.</span> <span m="3096480">The</span> <span m="3096580">probability</span> <span m="3102260">of</span> <span m="3102370">a</span> <span m="3102440">sample</span> <span m="3102870">point</span> <span m="3109680">is</span> <span m="3109960">the</span> <span m="3110070">product</span> <span m="3117980">of</span> <span m="3118080">the</span> <span m="3118180">probabilities</span> <span m="3128210">on</span> <span m="3128450">the</span> <span m="3128690">path</span> <span m="3129180">leading</span> <span m="3129590">to the</span> <span m="3129720">outcome,</span> <span m="3130250">the</span> <span m="3130330">path</span> <span m="3130700">in the tree</span> <span m="3140230">leading</span> <span m="3140580">to</span> <span m="3140690">the</span> <span m="3140830">outcome</span> <span m="3144580">or the</span> <span m="3144920">sample point.</span> <span m="3152360">OK.</span> </p>
<p><span m="3152970">So</span> <span m="3154250">what</span> <span m="3154660">this</span> <span m="3154840">says</span> <span m="3155170">is</span> <span m="3155280">just</span> <span m="3155470">really,</span> <span m="3155930">really</span> <span m="3156050">simple.</span> <span m="3158080">The</span> <span m="3158100">probability</span> <span m="3158650">of this</span> <span m="3158850">sample</span> <span m="3159190">point</span> <span m="3159660">is</span> <span m="3160870">1/3</span> <span m="3161960">times</span> <span m="3162050">1/3</span> <span m="3162770">times</span> <span m="3163410">1/2,</span> <span m="3165150">1/18.</span> <span m="3166620">The</span> <span m="3167080">probability</span> <span m="3167510">of this</span> <span m="3167700">sample</span> <span m="3168010">point</span> <span m="3169190">is</span> <span m="3169380">1/3</span> <span m="3171160">times</span> <span m="3171400">1/3</span> <span m="3172160">times</span> <span m="3172470">1/2,</span> <span m="3173740">1/18.</span> <span m="3175680">What's</span> <span m="3175890">the</span> <span m="3175980">probability</span> <span m="3176440">of</span> <span m="3176500">this</span> <span m="3176680">sample</span> <span m="3176980">point</span> <span m="3177240">going</span> <span m="3177360">to</span> <span m="3177410">be</span> <span m="3177590">here?</span> <span m="3180570">1/9.</span> <span m="3181130">I've got</span> <span m="3181440">1/3</span> <span m="3181840">times</span> <span m="3182060">1/3</span> <span m="3183080">times,</span> <span m="3183750">well, this g</span> <span m="3183850">was</span> <span m="3184250">1,</span> <span m="3185910">is</span> <span m="3186070">1/9.</span> <span m="3190030">The</span> <span m="3190050">next</span> <span m="3190380">one</span> <span m="3190570">here</span> <span m="3191770">is</span> <span m="3191940">1/3</span> <span m="3192430">times</span> <span m="3192620">1/3</span> <span m="3193110">times</span> <span m="3193380">1,</span> <span m="3194350">1/9.</span> <span m="3196630">Same</span> <span m="3196940">thing</span> <span m="3197100">here.</span> <span m="3198310">1/3</span> <span m="3198770">times</span> <span m="3198970">1/3</span> <span m="3199350">times</span> <span m="3199570">1</span> <span m="3199890">is</span> <span m="3200060">1/9.</span> <span m="3202720">Then</span> <span m="3202980">I</span> <span m="3203070">get</span> <span m="3203260">1/3</span> <span m="3204110">times</span> <span m="3204340">1/3</span> <span m="3204820">times</span> <span m="3205120">1/2</span> <span m="3206110">for</span> <span m="3206270">1/18.</span> <span m="3207920">And</span> <span m="3208150">I</span> <span m="3208210">can</span> <span m="3208460">keep</span> <span m="3208680">on</span> <span m="3208870">going.</span> <span m="3210940">This</span> <span m="3211360">would be</span> <span m="3211720">1/9.</span> <span m="3212970">This</span> <span m="3213640">one</span> <span m="3213930">has</span> <span m="3214090">a</span> <span m="3214280">1,</span> <span m="3214610">so</span> <span m="3214750">it's</span> <span m="3214860">a</span> <span m="3214910">1/9.</span> <span m="3218380">An</span> <span m="3218590">1/18</span> <span m="3220132">and</span> <span m="3221620">1/18.</span> <span m="3225100">OK.</span> </p>
<p><span m="3225660">Now,</span> <span m="3225750">that's</span> <span m="3225990">the</span> <span m="3226110">rule.</span> <span m="3226840">And</span> <span m="3226990">you'll</span> <span m="3227110">always</span> <span m="3227580">do</span> <span m="3227750">this.</span> <span m="3228900">You</span> <span m="3229050">can</span> <span m="3229240">sort</span> <span m="3229720">of</span> <span m="3229810">see</span> <span m="3230190">why</span> <span m="3230590">it</span> <span m="3230770">works.</span> <span m="3231680">You'll</span> <span m="3232090">understand</span> <span m="3232610">after</span> <span m="3232850">another</span> <span m="3233110">week</span> <span m="3233470">why,</span> <span m="3234170">for</span> <span m="3234310">sure,</span> <span m="3234590">why</span> <span m="3234760">it</span> <span m="3234880">works.</span> <span m="3235740">But</span> <span m="3235890">intuitively,</span> <span m="3238280">you know,</span> <span m="3238770">to</span> <span m="3239060">get</span> <span m="3239300">to</span> <span m="3239370">this</span> <span m="3239610">outcome,</span> <span m="3240080">these</span> <span m="3240320">are</span> <span m="3240340">the</span> <span m="3240450">choices</span> <span m="3240850">we</span> <span m="3240960">had</span> <span m="3241160">to</span> <span m="3241220">make.</span> <span m="3242170">First,</span> <span m="3242540">the</span> <span m="3242620">box</span> <span m="3242990">had</span> <span m="3243220">to</span> <span m="3243290">be--</span> <span m="3244340">sorry,</span> <span m="3244820">the</span> <span m="3244920">prize</span> <span m="3245300">had</span> <span m="3245400">to</span> <span m="3245440">be</span> <span m="3245550">in</span> <span m="3245600">box</span> <span m="3245920">1.</span> <span m="3246760">There's</span> <span m="3246910">a</span> <span m="3246980">one-in-three</span> <span m="3247520">chance</span> <span m="3247840">of</span> <span m="3247900">that</span> <span m="3248060">happening.</span> <span m="3249220">Now,</span> <span m="3249720">in</span> <span m="3249990">those</span> <span m="3250250">one-out-of-three</span> <span m="3251200">chances,</span> <span m="3253190">the</span> <span m="3253300">chance</span> <span m="3253710">the</span> <span m="3253810">player</span> <span m="3254190">also</span> <span m="3254540">picked</span> <span m="3254990">box</span> <span m="3255360">1,</span> <span m="3256250">1/3</span> <span m="3257130">of</span> <span m="3257210">the</span> <span m="3257320">time</span> <span m="3257890">of</span> <span m="3258040">1/3</span> <span m="3258700">of</span> <span m="3258750">the</span> <span m="3258870">time</span> <span m="3259200">that</span> <span m="3259370">happens.</span> <span m="3259990">All right?</span> <span m="3260950">1/3</span> <span m="3261490">of</span> <span m="3261530">the</span> <span m="3261640">time,</span> <span m="3261870">you're</span> <span m="3262020">here.</span> <span m="3262360">And of</span> <span m="3262420">that,</span> <span m="3263000">1/3</span> <span m="3263560">of</span> <span m="3263730">those</span> <span m="3264030">times,</span> <span m="3264760">you</span> <span m="3264880">get</span> <span m="3265050">to</span> <span m="3265140">here.</span> <span m="3266200">That's</span> <span m="3266360">1/3</span> <span m="3266640">times</span> <span m="3266830">1/3,</span> <span m="3267140">or</span> <span m="3267250">1/9.</span> <span m="3268094">And then,</span> <span m="3268516">1/2</span> <span m="3268940">of</span> <span m="3269170">those</span> <span m="3269520">times,</span> <span m="3270320">Carol</span> <span m="3270790">happen</span> <span m="3271130">to</span> <span m="3271250">reveal</span> <span m="3271610">box</span> <span m="3271890">2.</span> <span m="3272590">That's</span> <span m="3272910">1/18.</span> </p>
<p><span m="3275260">So do you</span> <span m="3275420">see</span> <span m="3275600">why</span> <span m="3276310">that</span> <span m="3276530">might</span> <span m="3276810">make</span> <span m="3277000">sense,</span> <span m="3277760">why</span> <span m="3277900">you're</span> <span m="3278010">multiplying</span> <span m="3278600">these?</span> <span m="3279580">Because</span> <span m="3279850">at</span> <span m="3279910">every</span> <span m="3280210">step,</span> <span m="3280540">you</span> <span m="3280630">have</span> <span m="3280720">a</span> <span m="3280770">chance</span> <span m="3281150">to</span> <span m="3281260">diverge.</span> <span m="3282400">And</span> <span m="3283690">the</span> <span m="3283790">probability</span> <span m="3284530">you take</span> <span m="3284780">the</span> <span m="3284860">step</span> <span m="3285150">that</span> <span m="3285230">we're</span> <span m="3285350">following</span> <span m="3285820">here--</span> <span m="3286955">say,</span> <span m="3287450">in</span> <span m="3287550">this</span> <span m="3287670">case,</span> <span m="3287850">it was</span> <span m="3287940">1/2--</span> <span m="3288540">that</span> <span m="3288760">cuts</span> <span m="3289070">down</span> <span m="3289610">the</span> <span m="3289710">chance</span> <span m="3290100">you</span> <span m="3290210">land</span> <span m="3290530">here</span> <span m="3291670">by</span> <span m="3291960">that</span> <span m="3292430">probability,</span> <span m="3292960">1/2.</span> </p>
<p><span m="3295320">Any</span> <span m="3295470">questions</span> <span m="3295850">about that?</span> <span m="3298778">All right.</span> <span m="3299270">For</span> <span m="3299400">now,</span> <span m="3299600">you</span> <span m="3299710">can</span> <span m="3299840">take</span> <span m="3300050">it on</span> <span m="3300210">faith.</span> <span m="3300590">We'll</span> <span m="3300750">see</span> <span m="3302230">why</span> <span m="3302630">after</span> <span m="3302820">we</span> <span m="3302910">do</span> <span m="3303070">a</span> <span m="3303130">few</span> <span m="3303300">more</span> <span m="3303530">of</span> <span m="3303630">these</span> <span m="3304190">over</span> <span m="3304330">the</span> <span m="3304390">next</span> <span m="3304620">week</span> <span m="3304800">or</span> <span m="3304910">so.</span> <span m="3307760">Any</span> <span m="3307930">questions</span> <span m="3308630">there?</span> <span m="3309280">Is that OK?</span> <span m="3312160">OK.</span> <span m="3312600">Oh,</span> <span m="3312720">yeah,</span> <span m="3313080">all right?</span> </p>
<p><span m="3314487">AUDIENCE :</span> <span m="3314956">[INAUDIBLE]</span> </p>
<p><span m="3319180">PROFESSOR: On this</span> <span m="3319400">one?</span> <span m="3320880">Only</span> <span m="3321130">one</span> <span m="3321350">choice.</span> <span m="3322160">So</span> <span m="3322360">it</span> <span m="3322620">happens</span> <span m="3322875">with</span> <span m="3323130">probability</span> <span m="3323610">1.</span> <span m="3327220">Any</span> <span m="3327410">other</span> <span m="3327610">questions?</span> <span m="3331459">OK.</span> <span m="3332860">All</span> <span m="3333030">right.</span> <span m="3333970">So</span> <span m="3334170">now</span> <span m="3334610">we</span> <span m="3334720">can</span> <span m="3334870">figure</span> <span m="3335260">out</span> <span m="3335480">the</span> <span m="3335580">probability</span> <span m="3336440">of</span> <span m="3336590">winning.</span> <span m="3338780">Well,</span> <span m="3339770">the</span> <span m="3340370">probability</span> <span m="3340830">of</span> <span m="3340900">winning</span> <span m="3341260">is</span> <span m="3341340">we</span> <span m="3341430">have</span> <span m="3341570">1/9,</span> <span m="3342210">1/9,</span> <span m="3342790">there's</span> <span m="3343010">3</span> <span m="3343320">of them,</span> <span m="3343970">4</span> <span m="3344320">of them,</span> <span m="3345600">5,</span> <span m="3346000">6.</span> <span m="3346910">All right?</span> <span m="3347300">So</span> <span m="3347730">the</span> <span m="3347850">probability</span> <span m="3348420">of</span> <span m="3348490">winning</span> <span m="3348810">here</span> <span m="3349930">is</span> <span m="3350180">6</span> <span m="3351190">times</span> <span m="3351640">1/9.</span> </p>
<p><span m="3376190">Because</span> <span m="3376360">you</span> <span m="3376470">just</span> <span m="3376680">sum</span> <span m="3376940">up</span> <span m="3377070">the</span> <span m="3377170">probabilities</span> <span m="3378120">of</span> <span m="3378320">the</span> <span m="3378480">winning</span> <span m="3378730">sample</span> <span m="3379110">points.</span> <span m="3380240">There's</span> <span m="3380330">six</span> <span m="3380760">of</span> <span m="3380850">winning.</span> <span m="3381310">They're</span> <span m="3381550">each</span> <span m="3381780">1/9.</span> <span m="3383230">And</span> <span m="3383680">that's</span> <span m="3384130">2/3.</span> <span m="3386380">OK?</span> <span m="3386750">So</span> <span m="3388100">Marilyn</span> <span m="3388410">was</span> <span m="3388610">right.</span> <span m="3389110">There's</span> <span m="3389270">a</span> <span m="3389360">2/3</span> <span m="3389830">chance</span> <span m="3390120">of</span> <span m="3390210">winning</span> <span m="3390670">if</span> <span m="3390840">you</span> <span m="3390950">switch.</span> <span m="3391660">And</span> <span m="3391810">now,</span> <span m="3391970">this</span> <span m="3392270">is</span> <span m="3392400">a</span> <span m="3392460">formal</span> <span m="3392890">argument</span> <span m="3393900">that</span> <span m="3394090">gives</span> <span m="3394230">you</span> <span m="3394340">that</span> <span m="3394550">answer.</span> <span m="3395630">The</span> <span m="3395760">nice</span> <span m="3395980">thing</span> <span m="3396120">is</span> <span m="3396230">it</span> <span m="3396290">works</span> <span m="3396590">for</span> <span m="3397080">pretty</span> <span m="3397220">much</span> <span m="3397430">every</span> <span m="3397670">problem</span> <span m="3397980">you ever</span> <span m="3398190">think</span> <span m="3398410">about.</span> </p>
<p><span m="3399880">Now,</span> <span m="3400270">in</span> <span m="3400410">general,</span> <span m="3400920">if</span> <span m="3401000">you</span> <span m="3401110">want to</span> <span m="3401360">know</span> <span m="3401430">the</span> <span m="3401540">probability</span> <span m="3402180">of</span> <span m="3402360">any</span> <span m="3402700">event,</span> <span m="3403920">you</span> <span m="3404050">can</span> <span m="3404210">follow</span> <span m="3404510">this</span> <span m="3404900">same</span> <span m="3405200">method.</span> <span m="3405540">Let</span> <span m="3405640">me</span> <span m="3405720">define</span> <span m="3406150">what</span> <span m="3406260">an</span> <span m="3406360">event</span> <span m="3406740">is.</span> <span m="3413085">An event</span> <span m="3416410">is</span> <span m="3416720">simply</span> <span m="3417170">a</span> <span m="3417240">subset</span> <span m="3418200">of</span> <span m="3418500">outcomes,</span> <span m="3418880">a</span> <span m="3418930">subset</span> <span m="3419290">of</span> <span m="3419330">the</span> <span m="3419420">sample</span> <span m="3419690">space.</span> <span m="3430040">For</span> <span m="3430130">example,</span> <span m="3431230">we</span> <span m="3431380">could</span> <span m="3431510">define</span> <span m="3431940">the</span> <span m="3432070">event</span> <span m="3432400">of</span> <span m="3432500">losing.</span> <span m="3434530">Let's</span> <span m="3434750">call</span> <span m="3435070">EL,</span> <span m="3436310">E</span> <span m="3436650">sub-L,</span> <span m="3436990">to be</span> <span m="3437130">the</span> <span m="3437290">event</span> <span m="3439690">that</span> <span m="3440080">player</span> <span m="3440460">loses,</span> <span m="3440840">the</span> <span m="3441220">player</span> <span m="3441470">loses,</span> <span m="3446120">with</span> <span m="3446320">the</span> <span m="3446410">switch</span> <span m="3446760">strategy.</span> </p>
<p><span m="3455630">All</span> <span m="3455750">right?</span> <span m="3455890">Now,</span> <span m="3456080">to</span> <span m="3456470">compute</span> <span m="3456840">the</span> <span m="3456930">probability</span> <span m="3457530">of</span> <span m="3457590">an</span> <span m="3457680">event,</span> <span m="3458110">we</span> <span m="3458210">just</span> <span m="3458620">add</span> <span m="3459940">the</span> <span m="3460060">probabilities</span> <span m="3460790">of</span> <span m="3460850">the</span> <span m="3460990">outcomes</span> <span m="3461640">in</span> <span m="3461810">that</span> <span m="3462050">set.</span> <span m="3464110">So</span> <span m="3464300">let's</span> <span m="3465457">get</span> <span m="3465954">that</span> <span m="3466948">down.</span> <span m="3485840">All</span> <span m="3485920">right,</span> <span m="3486220">so</span> <span m="3489210">formally,</span> <span m="3491140">the</span> <span m="3491640">probability</span> <span m="3499577">that</span> <span m="3500070">an</span> <span m="3500170">event</span> <span m="3500520">occurs</span> <span m="3511190">is</span> <span m="3511350">simply</span> <span m="3511770">the</span> <span m="3511890">sum,</span> <span m="3515694">we'll</span> <span m="3516150">call</span> <span m="3516360">an</span> <span m="3516450">event,</span> <span m="3516805">e,</span> <span m="3517590">let's</span> <span m="3517920">say,</span> <span m="3518500">event,</span> <span m="3518820">e.</span> <span m="3519590">It's a</span> <span m="3520000">subset</span> <span m="3520380">of</span> <span m="3520440">the</span> <span m="3520520">sample</span> <span m="3520920">space.</span> <span m="3521380">The</span> <span m="3521470">probability</span> <span m="3521730">that</span> <span m="3522070">e</span> <span m="3522280">occurs</span> <span m="3523340">is</span> <span m="3523510">the</span> <span m="3523610">sum</span> <span m="3524710">over</span> <span m="3525000">the</span> <span m="3525480">sample</span> <span m="3525840">points</span> <span m="3526150">in</span> <span m="3526210">e</span> <span m="3527123">of</span> <span m="3527586">the</span> <span m="3528050">probability</span> <span m="3529260">of</span> <span m="3529380">those</span> <span m="3529550">sample</span> <span m="3529920">points.</span> <span m="3531650">So</span> <span m="3531800">for</span> <span m="3531930">example,</span> <span m="3534090">the</span> <span m="3534100">probability</span> <span m="3534960">of</span> <span m="3535030">the</span> <span m="3535170">event,</span> <span m="3535700">EL,</span> <span m="3536280">where</span> <span m="3536410">the</span> <span m="3536460">player</span> <span m="3536800">loses,</span> <span m="3539650">well,</span> <span m="3539930">there's</span> <span m="3540850">six</span> <span m="3541210">sample points</span> <span m="3541500">where</span> <span m="3541610">the</span> <span m="3541710">player</span> <span m="3542020">loses,</span> <span m="3543250">each</span> <span m="3543560">with the</span> <span m="3543700">probability</span> <span m="3544190">1/18.</span> <span m="3547760">So</span> <span m="3547950">it's</span> <span m="3548050">1/3</span> <span m="3549901">is</span> <span m="3550330">the</span> <span m="3550480">probability</span> <span m="3550950">the player</span> <span m="3551240">loses</span> <span m="3551610">on</span> <span m="3551700">the</span> <span m="3551780">switch</span> <span m="3552040">strategy.</span> <span m="3553140">And</span> <span m="3553320">that's</span> <span m="3553560">not</span> <span m="3553730">surprising</span> <span m="3554260">because</span> <span m="3555720">you</span> <span m="3555890">either</span> <span m="3556060">win</span> <span m="3556250">or</span> <span m="3556270">you</span> <span m="3556440">lose.</span> <span m="3556880">The</span> <span m="3556970">probabilities</span> <span m="3557590">have</span> <span m="3557760">to</span> <span m="3557890">add</span> <span m="3558080">to</span> <span m="3558160">1.</span> <span m="3558970">So</span> <span m="3558990">if</span> <span m="3559070">won with</span> <span m="3559570">probability</span> <span m="3560010">2/3,</span> <span m="3561110">you</span> <span m="3561280">lose</span> <span m="3561970">with</span> <span m="3562080">probability</span> <span m="3562520">1/3.</span> <span m="3565570">OK?</span> <span m="3568050">All right.</span> </p>
<p><span m="3568190">So</span> <span m="3568240">Marilyn</span> <span m="3568610">was</span> <span m="3568780">right,</span> <span m="3569110">much</span> <span m="3569330">better</span> <span m="3569530">to</span> <span m="3569620">switch.</span> <span m="3571860">We'll</span> <span m="3571990">do</span> <span m="3572080">a</span> <span m="3572130">bunch</span> <span m="3572340">more</span> <span m="3572490">examples</span> <span m="3572930">with</span> <span m="3573000">Monty</span> <span m="3573330">Hall</span> <span m="3573570">tomorrow.</span> <span m="3574810">Are</span> <span m="3574860">there</span> <span m="3574990">any</span> <span m="3575160">other</span> <span m="3576230">questions</span> <span m="3576720">on</span> <span m="3576820">Monty</span> <span m="3577130">Hall</span> <span m="3578400">before</span> <span m="3578825">I</span> <span m="3579250">change</span> <span m="3579630">gears?</span> <span m="3580590">Yeah?</span> </p>
<p><span m="3582572">AUDIENCE: So you showed that</span> <span m="3583010">when you</span> <span m="3583450">switch,</span> <span m="3584480">you</span> <span m="3585110">are</span> <span m="3585740">more</span> <span m="3586220">likely</span> <span m="3586700">to win than</span> <span m="3587180">to lose.</span> <span m="3588140">But you didn't</span> <span m="3588620">show that</span> <span m="3589100">that's not also true</span> <span m="3589580">when you don't switch.</span> </p>
<p><span m="3591020">PROFESSOR: Ah</span> <span m="3592470">good question.</span> <span m="3592810">What happens if you're</span> <span m="3593150">playing</span> <span m="3593430">the</span> <span m="3593600">stay</span> <span m="3594030">strategy.</span> <span m="3598900">Right,</span> <span m="3599960">maybe</span> <span m="3600270">you</span> <span m="3600410">win</span> <span m="3600780">2/3</span> <span m="3601180">of</span> <span m="3601220">the</span> <span m="3601310">time</span> <span m="3601570">when</span> <span m="3601680">you</span> <span m="3601770">stay.</span> <span m="3603060">Maybe.</span> <span m="3604190">Maybe</span> <span m="3604370">not.</span> <span m="3605730">Right?</span> <span m="3606090">Because</span> <span m="3607450">if</span> <span m="3607630">I</span> <span m="3610090">win</span> <span m="3610840">when</span> <span m="3611010">I</span> <span m="3611130">switch,</span> <span m="3612820">what</span> <span m="3613090">would</span> <span m="3613210">have</span> <span m="3613310">happened</span> <span m="3613760">if</span> <span m="3613910">I'd</span> <span m="3614080">stayed?</span> <span m="3617680">I'd have</span> <span m="3617840">lost.</span> <span m="3619570">Because</span> <span m="3619780">if</span> <span m="3619870">I</span> <span m="3619960">win</span> <span m="3620360">when</span> <span m="3620490">I</span> <span m="3620580">switch,</span> <span m="3621000">it</span> <span m="3621080">means</span> <span m="3621330">the</span> <span m="3621410">prize is</span> <span m="3621900">over</span> <span m="3622080">there.</span> <span m="3622380">So if</span> <span m="3622830">I stay,</span> <span m="3623050">I'm</span> <span m="3623160">going</span> <span m="3623260">to</span> <span m="3623320">lose.</span> <span m="3624990">Which</span> <span m="3625180">means</span> <span m="3626240">that</span> <span m="3626350">if</span> <span m="3626430">I</span> <span m="3626520">do</span> <span m="3626630">the</span> <span m="3626750">stay</span> <span m="3627100">strategy,</span> <span m="3627680">I'm</span> <span m="3627760">going</span> <span m="3627860">to</span> <span m="3627930">lose</span> <span m="3628280">2/3</span> <span m="3628730">of</span> <span m="3628780">the</span> <span m="3628870">time.</span> <span m="3631990">So</span> <span m="3632100">you're</span> <span m="3632200">right.</span> <span m="3632410">I</span> <span m="3632480">didn't</span> <span m="3632740">cover</span> <span m="3632980">that.</span> <span m="3633750">But</span> <span m="3633960">we</span> <span m="3634090">can.</span> <span m="3634420">It's</span> <span m="3634580">not</span> <span m="3634830">too</span> <span m="3634990">hard</span> <span m="3635220">in</span> <span m="3635280">this</span> <span m="3635460">case</span> <span m="3636400">because</span> <span m="3637540">the</span> <span m="3637630">probability</span> <span m="3638160">of</span> <span m="3638240">winning</span> <span m="3638690">on</span> <span m="3638810">a</span> <span m="3638860">switch</span> <span m="3640560">equals</span> <span m="3640910">the</span> <span m="3641000">probability</span> <span m="3641540">of</span> <span m="3641620">losing</span> <span m="3642120">on a stick.</span> </p>
<p><span m="3645760">So</span> <span m="3645990">the</span> <span m="3646090">probability</span> <span m="3647460">that you</span> <span m="3647760">win</span> <span m="3649010">with</span> <span m="3649200">switch</span> <span m="3652280">equals</span> <span m="3652940">the</span> <span m="3653040">probability</span> <span m="3653740">you</span> <span m="3653870">lose</span> <span m="3655860">with</span> <span m="3656210">a</span> <span m="3656290">stick</span> <span m="3658300">strategy.</span> <span m="3662560">All</span> <span m="3662750">right?</span> <span m="3663080">And</span> <span m="3663260">they're</span> <span m="3663420">both</span> <span m="3663980">2/3,</span> <span m="3665200">because</span> <span m="3665430">they're</span> <span m="3665540">the</span> <span m="3665650">same</span> <span m="3666080">and this</span> <span m="3666300">is</span> <span m="3666390">2/3.</span> <span m="3667950">So</span> <span m="3668030">in</span> <span m="3668110">fact,</span> <span m="3668460">what's,</span> <span m="3668770">then,</span> <span m="3669190">the</span> <span m="3669280">chance</span> <span m="3669600">you</span> <span m="3669740">win</span> <span m="3670070">with</span> <span m="3670210">a</span> <span m="3670260">stick</span> <span m="3670510">strategy?</span> <span m="3673350">1/3.</span> <span m="3674990">And</span> <span m="3675190">that's</span> <span m="3675380">sort</span> <span m="3675570">of</span> <span m="3676260">you know, where</span> <span m="3676420">Marilyn</span> <span m="3676710">was</span> <span m="3676860">coming</span> <span m="3677150">from.</span> <span m="3677410">Because</span> <span m="3677760">she</span> <span m="3677870">said,</span> <span m="3678030">well,</span> <span m="3678700">if</span> <span m="3678830">you</span> <span m="3678990">pick</span> <span m="3679200">the</span> <span m="3679290">box</span> <span m="3679640">right</span> <span m="3679920">the</span> <span m="3680000">first</span> <span m="3680300">time,</span> <span m="3680720">which</span> <span m="3680750">is</span> <span m="3680830">what</span> <span m="3680990">you</span> <span m="3681080">need</span> <span m="3681260">to</span> <span m="3681320">get</span> <span m="3681620">to</span> <span m="3681710">win</span> <span m="3681910">with</span> <span m="3682030">a</span> <span m="3682070">stick</span> <span m="3682310">strategy,</span> <span m="3683870">the</span> <span m="3684110">chance</span> <span m="3684380">you</span> <span m="3684480">pick</span> <span m="3684690">the</span> <span m="3684780">right</span> <span m="3684990">box</span> <span m="3685360">is</span> <span m="3685530">one-in-three.</span> <span m="3686830">And</span> <span m="3686980">showing,</span> <span m="3687270">you know,</span> <span m="3687800">a</span> <span m="3687860">goat</span> <span m="3688120">over</span> <span m="3688300">here</span> <span m="3688470">didn't</span> <span m="3688670">matter.</span> <span m="3689060">It's</span> <span m="3689770">still</span> <span m="3690020">one-in-three.</span> </p>
<p><span m="3692200">Now</span> <span m="3692410">that's</span> <span m="3692770">gets a--</span> <span m="3693170">that's</span> <span m="3693830">intuition,</span> <span m="3694330">but it</span> <span m="3694450">can</span> <span m="3694700">get</span> <span m="3694830">you</span> <span m="3694950">trouble,</span> <span m="3695240">because</span> <span m="3697610">say</span> <span m="3698260">you</span> <span m="3698370">picked</span> <span m="3698600">this</span> <span m="3698790">box and</span> <span m="3699360">Carol</span> <span m="3699680">showed</span> <span m="3699980">you</span> <span m="3700080">both</span> <span m="3700630">of</span> <span m="3700760">these</span> <span m="3701000">boxes,</span> <span m="3701450">and</span> <span m="3701510">they</span> <span m="3701590">had</span> <span m="3701800">pocket</span> <span m="3702110">protectors.</span> <span m="3703920">Then</span> <span m="3704120">what</span> <span m="3704260">do</span> <span m="3704300">you</span> <span m="3704390">know</span> <span m="3704570">about</span> <span m="3704720">the</span> <span m="3704800">probability</span> <span m="3705310">you</span> <span m="3705780">won</span> <span m="3706100">with</span> <span m="3706150">a</span> <span m="3706390">stick?</span> <span m="3707750">It's</span> <span m="3707920">1.</span> <span m="3708710">Ooh,</span> <span m="3709020">so</span> <span m="3709100">things</span> <span m="3709310">did</span> <span m="3709490">change</span> <span m="3709840">there.</span> <span m="3711026">All right?</span> <span m="3711400">So you</span> <span m="3711690">have to be</span> <span m="3711820">a</span> <span m="3711930">little</span> <span m="3712210">careful</span> <span m="3712630">about</span> <span m="3713080">how</span> <span m="3713290">you</span> <span m="3713460">reason,</span> <span m="3714375">OK?</span> <span m="3715740">You've</span> <span m="3715870">got</span> <span m="3716050">to</span> <span m="3716100">do</span> <span m="3716220">this.</span> <span m="3718300">Stay</span> <span m="3718480">with</span> <span m="3718600">that</span> <span m="3718960">and you'll</span> <span m="3719170">always</span> <span m="3719590">be</span> <span m="3719730">fine</span> <span m="3721140">for</span> <span m="3721230">your</span> <span m="3721360">strategy.</span> <span m="3723630">So</span> <span m="3724260">really</span> <span m="3724550">the key</span> <span m="3724860">is</span> <span m="3725000">to</span> <span m="3725090">resist</span> <span m="3725600">the</span> <span m="3725700">temptation</span> <span m="3727160">to</span> <span m="3727280">just</span> <span m="3727470">sort</span> <span m="3727670">of</span> <span m="3728310">use</span> <span m="3728610">your</span> <span m="3728770">intuition</span> <span m="3729290">to</span> <span m="3729360">think</span> <span m="3729590">about</span> <span m="3729880">it.</span> <span m="3732224">All right.</span> </p>
<p><span m="3732890">Now,</span> <span m="3733530">this</span> <span m="3733700">is</span> <span m="3733810">a</span> <span m="3733880">pretty</span> <span m="3734130">simple</span> <span m="3734450">example.</span> <span m="3737530">So</span> <span m="3737700">now</span> <span m="3737830">I'm</span> <span m="3737940">going</span> <span m="3738050">to</span> <span m="3738100">do</span> <span m="3738250">one</span> <span m="3738660">that's</span> <span m="3739270">a</span> <span m="3739340">little</span> <span m="3739550">more</span> <span m="3739830">interesting</span> <span m="3741270">and</span> <span m="3741540">a</span> <span m="3741600">little</span> <span m="3742030">more</span> <span m="3742230">devious</span> <span m="3742800">in</span> <span m="3742880">a</span> <span m="3742940">sense.</span> <span m="3744200">So</span> <span m="3744700">I</span> <span m="3744780">need</span> <span m="3744940">a</span> <span m="3744970">volunteer.</span> <span m="3745540">We're</span> <span m="3745620">going</span> <span m="3745700">to</span> <span m="3745740">do</span> <span m="3745840">a</span> <span m="3745900">gambling</span> <span m="3746370">game.</span> <span m="3747610">So</span> <span m="3747850">I</span> <span m="3747930">need</span> <span m="3748090">a</span> <span m="3748120">volunteer.</span> <span m="3749700">All</span> <span m="3750010">right.</span> <span m="3750570">Now,</span> <span m="3750760">he's</span> <span m="3751150">raised</span> <span m="3751450">his</span> <span m="3751510">hands</span> <span m="3751820">before</span> <span m="3752110">I even</span> <span m="3752330">said</span> <span m="3752460">what</span> <span m="3752590">the</span> <span m="3752680">criteria</span> <span m="3753390">are.</span> <span m="3754750">First,</span> <span m="3755110">you</span> <span m="3755180">gotta</span> <span m="3755650">be</span> <span m="3755760">confident</span> <span m="3756230">you</span> <span m="3756300">understand</span> <span m="3756720">probability.</span> <span m="3760570">Well,</span> <span m="3760780">you</span> <span m="3760890">can</span> <span m="3761090">fake</span> <span m="3761300">it.</span> <span m="3761470">But</span> <span m="3761520">second,</span> <span m="3761880">and</span> <span m="3761950">more</span> <span m="3762120">importantly,</span> <span m="3763050">you've</span> <span m="3763170">got</span> <span m="3763290">to</span> <span m="3763350">have</span> <span m="3763520">some</span> <span m="3763670">money</span> <span m="3764040">in your</span> <span m="3764100">product</span> <span m="3765680">or</span> <span m="3766020">you've</span> <span m="3766070">got</span> <span m="3766200">to</span> <span m="3766250">borrow</span> <span m="3766620">some</span> <span m="3766850">on</span> <span m="3766900">the</span> <span m="3766990">way</span> <span m="3767160">down.</span> <span m="3768560">You got</span> <span m="3768640">a</span> <span m="3768680">few</span> <span m="3768850">bucks</span> <span m="3769180">there?</span> </p>
<p><span m="3770185">AUDIENCE: Yes.</span> </p>
<p><span m="3771400">PROFESSOR: All right,</span> <span m="3771500">come on down.</span> <span m="3772191">That's</span> <span m="3772672">really</span> <span m="3773153">all that matters,</span> <span m="3773634">you know.</span> <span m="3783735">OK,</span> <span m="3784220">now, in</span> <span m="3784330">this</span> <span m="3784550">game,</span> <span m="3784870">there's</span> <span m="3785140">three</span> <span m="3785500">dice.</span> <span m="3788370">Well,</span> <span m="3788610">that's</span> <span m="3788790">'cause</span> <span m="3788910">you're</span> <span m="3789050">smart.</span> <span m="3790700">OK,</span> <span m="3791800">so</span> <span m="3792100">let</span> <span m="3792160">me</span> <span m="3792260">show</span> <span m="3792470">you</span> <span m="3792570">the</span> <span m="3792670">dice</span> <span m="3793010">in</span> <span m="3793080">this</span> <span m="3793260">game.</span> <span m="3800980">And</span> <span m="3801140">these,</span> <span m="3801380">of</span> <span m="3801460">course,</span> <span m="3801830">are</span> <span m="3801880">not</span> <span m="3802240">normal</span> <span m="3802610">dice.</span> <span m="3818290">All</span> <span m="3818390">right?</span> <span m="3818570">So</span> <span m="3819880">dice A</span> <span m="3820290">has</span> <span m="3821440">a</span> <span m="3821720">2,</span> <span m="3822000">a</span> <span m="3822060">6,</span> <span m="3822315">and</span> <span m="3822570">a</span> <span m="3822770">7,</span> <span m="3823750">and</span> <span m="3823890">they're</span> <span m="3823980">the</span> <span m="3824110">same</span> <span m="3824530">on</span> <span m="3824670">the</span> <span m="3824760">reverse</span> <span m="3825110">side.</span> <span m="3825390">So</span> <span m="3825470">it's</span> <span m="3825570">a</span> <span m="3825620">six-sided</span> <span m="3826190">die,</span> <span m="3826880">and</span> <span m="3827140">it's</span> <span m="3827310">fair.</span> <span m="3827630">When</span> <span m="3827730">you</span> <span m="3827880">roll</span> <span m="3828350">it,</span> <span m="3828940">the</span> <span m="3829070">probability</span> <span m="3829820">A</span> <span m="3829990">comes</span> <span m="3830290">up</span> <span m="3830460">2</span> <span m="3830770">is</span> <span m="3830860">going</span> <span m="3830960">to</span> <span m="3831000">be</span> <span m="3831100">1/3.</span> <span m="3831970">All right?</span> <span m="3832490">Same</span> <span m="3833100">for</span> <span m="3833220">6</span> <span m="3833500">and</span> <span m="3833580">for</span> <span m="3833710">7.</span> <span m="3835640">Ah,</span> <span m="3836350">we're</span> <span m="3836480">just</span> <span m="3836720">getting</span> <span m="3836910">started.</span> </p>
<p><span m="3838850">Dice</span> <span m="3838940">B</span> <span m="3839430">has</span> <span m="3839630">a</span> <span m="3839710">1,</span> <span m="3840010">5,</span> <span m="3840380">and</span> <span m="3840590">a 9</span> <span m="3841890">and</span> <span m="3842160">dice</span> <span m="3842260">C</span> <span m="3842830">has</span> <span m="3843320">a 3,</span> <span m="3844500">4,</span> <span m="3845150">and</span> <span m="3845290">an</span> <span m="3845500">8.</span> <span m="3847650">So</span> <span m="3847760">I</span> <span m="3847810">got--</span> <span m="3848240">they're</span> <span m="3848380">all</span> <span m="3848610">different</span> <span m="3848890">numbers,</span> <span m="3849260">the</span> <span m="3849320">numbers</span> <span m="3849620">from 1</span> <span m="3849900">to</span> <span m="3849970">9.</span> <span m="3851760">So</span> <span m="3852050">just</span> <span m="3852260">so</span> <span m="3852380">we're</span> <span m="3852560">really</span> <span m="3852860">clear,</span> <span m="3854160">the</span> <span m="3854230">probability</span> <span m="3855190">that</span> <span m="3855700">A</span> <span m="3856010">comes</span> <span m="3856340">up</span> <span m="3856590">3</span> <span m="3858000">is</span> <span m="3858320">1/3.</span> <span m="3860100">The</span> <span m="3860220">same</span> <span m="3860680">for</span> <span m="3860840">the</span> <span m="3860930">probability</span> <span m="3861730">A</span> <span m="3862520">comes</span> <span m="3862770">up</span> <span m="3863420">6</span> <span m="3864310">is</span> <span m="3864510">1/3</span> <span m="3865430">and</span> <span m="3865730">so</span> <span m="3865970">on.</span> <span m="3868394">Ah,</span> <span m="3868850">yes</span> <span m="3869270">I</span> <span m="3869310">do</span> <span m="3869480">mean</span> <span m="3869670">2.</span> <span m="3870405">Good.</span> <span m="3871110">All right, good,</span> <span m="3871340">you're</span> <span m="3871530">paying</span> <span m="3871670">attention.</span> <span m="3871955">That's</span> <span m="3872240">good.</span> <span m="3872480">There's</span> <span m="3873010">some</span> <span m="3873365">hope.</span> <span m="3875070">All right?</span> </p>
<p><span m="3875520">Now,</span> <span m="3875870">here's</span> <span m="3876150">the</span> <span m="3876240">game.</span> <span m="3876990">We're</span> <span m="3877190">each</span> <span m="3877360">going</span> <span m="3877460">to</span> <span m="3877510">pick</span> <span m="3877690">a</span> <span m="3877780">die.</span> <span m="3878610">And</span> <span m="3878750">we're</span> <span m="3878850">going</span> <span m="3878950">to</span> <span m="3879060">roll</span> <span m="3879390">our</span> <span m="3879500">die.</span> <span m="3880760">And</span> <span m="3881230">the</span> <span m="3881290">higher</span> <span m="3881670">die</span> <span m="3881990">wins.</span> <span m="3882580">We're going to</span> <span m="3882700">roll</span> <span m="3882950">them once.</span> <span m="3883140">The</span> <span m="3883260">higher</span> <span m="3883550">one</span> <span m="3883740">wins.</span> <span m="3884600">And</span> <span m="3884800">the</span> <span m="3884890">loser</span> <span m="3885330">pays</span> <span m="3885650">the</span> <span m="3885750">winner</span> <span m="3886290">$1.</span> <span m="3887420">And</span> <span m="3888080">no</span> <span m="3888270">tears,</span> <span m="3888710">OK?</span> <span m="3890560">You got</span> <span m="3890870">that?</span> <span m="3891450">That's good.</span> <span m="3891750">He's already</span> <span m="3891840">trying</span> <span m="3891970">to</span> <span m="3892050">figure</span> <span m="3892230">out</span> <span m="3892290">which</span> <span m="3892440">one</span> <span m="3892560">to</span> <span m="3892640">pick.</span> <span m="3893730">So</span> <span m="3893910">let's</span> <span m="3894100">put</span> <span m="3894210">your</span> <span m="3894310">dollar</span> <span m="3894600">on</span> <span m="3894690">the</span> <span m="3894770">table.</span> <span m="3897290">Here</span> <span m="3897743">we go.</span> </p>
<p><span m="3899780">AUDIENCE: I have a $5.</span> <span m="3899980">Do you</span> <span m="3900406">have change?</span> </p>
<p><span m="3901260">PROFESSOR: I might</span> <span m="3901460">have</span> <span m="3901610">change.</span> <span m="3903460">All right.</span> <span m="3904150">Let's</span> <span m="3904210">see</span> <span m="3904600">here.</span> <span m="3906630">Yes,</span> <span m="3906990">I</span> <span m="3907060">do</span> <span m="3907200">have</span> <span m="3907350">change.</span> </p>
<p><span m="3907750">AUDIENCE: Oh, I do have a dollar.</span> </p>
<p><span m="3908150">PROFESSOR: You have</span> <span m="3908250">a</span> <span m="3908350">dollar.</span> <span m="3908810">All right.</span> <span m="3909962">Well,</span> <span m="3910300">for</span> <span m="3910420">now</span> <span m="3910660">anyway.</span> <span m="3911020">Let's</span> <span m="3911230">put</span> <span m="3911370">the</span> <span m="3911450">dollar</span> <span m="3911780">there</span> <span m="3913640">on</span> <span m="3913780">the</span> <span m="3913860">table.</span> </p>
<p><span m="3915330">AUDIENCE: It's</span> <span m="3915810">going to be a $20 next.</span> </p>
<p><span m="3916290">PROFESSOR: We're</span> <span m="3916400">just</span> <span m="3916570">getting</span> <span m="3916760">warmed</span> <span m="3917050">up</span> <span m="3917190">here.</span> <span m="3917480">OK,</span> <span m="3917810">now,</span> <span m="3919040">there</span> <span m="3919350">is--</span> <span m="3919600">what's</span> <span m="3919740">your</span> <span m="3919830">name,</span> <span m="3920020">by</span> <span m="3920150">the</span> <span m="3920250">way?</span> </p>
<p><span m="3920710">AUDIENCE: Will.</span> </p>
<p><span m="3920730">PROFESSOR: Will.</span> <span m="3921660">OK,</span> <span m="3922350">nice</span> <span m="3922540">to</span> <span m="3922620">meet</span> <span m="3922720">you,</span> <span m="3922850">Will.</span> <span m="3923890">There</span> <span m="3924370">we go.</span> <span m="3925030">All right.</span> <span m="3925290">Now,</span> <span m="3925960">there</span> <span m="3926170">is</span> <span m="3926340">one</span> <span m="3926720">problem</span> <span m="3927140">here.</span> <span m="3927830">I</span> <span m="3927950">don't</span> <span m="3928250">really</span> <span m="3928810">have</span> <span m="3929090">those</span> <span m="3929340">dice.</span> <span m="3930840">So</span> <span m="3931090">we're</span> <span m="3931190">going</span> <span m="3931270">to</span> <span m="3931310">have</span> <span m="3931470">to</span> <span m="3931560">do</span> <span m="3931680">a</span> <span m="3931730">mathematical</span> <span m="3932360">version</span> <span m="3932720">of</span> <span m="3932790">this</span> <span m="3933100">bet.</span> </p>
<p><span m="3934022">AUDIENCE: You're</span> <span m="3934454">always going to win,</span> <span m="3934886">then.</span> </p>
<p><span m="3935320">PROFESSOR: So</span> <span m="3936090">what</span> <span m="3936290">we're</span> <span m="3936400">going</span> <span m="3936510">to</span> <span m="3936610">do</span> <span m="3937590">is</span> <span m="3937980">pretend</span> <span m="3938450">that</span> <span m="3938530">we</span> <span m="3938650">did</span> <span m="3938810">a</span> <span m="3938890">roll,</span> <span m="3939510">head-to-head,</span> <span m="3940710">and</span> <span m="3941370">whichever</span> <span m="3941710">one</span> <span m="3941890">is</span> <span m="3941980">more</span> <span m="3942210">likely</span> <span m="3942720">to</span> <span m="3942800">win,</span> <span m="3943770">we're</span> <span m="3943910">going</span> <span m="3944010">to</span> <span m="3944050">assume</span> <span m="3944310">the</span> <span m="3944400">roll</span> <span m="3944600">came</span> <span m="3944820">out</span> <span m="3944970">that</span> <span m="3945130">way,</span> <span m="3946170">all</span> <span m="3946280">right?</span> <span m="3947460">Now,</span> <span m="3948710">I</span> <span m="3948820">know</span> <span m="3948950">you've</span> <span m="3949100">been</span> <span m="3949210">studying</span> <span m="3949660">those</span> <span m="3949960">three</span> <span m="3950820">guys</span> <span m="3951160">there,</span> <span m="3951330">those</span> <span m="3951500">three</span> <span m="3951680">dice.</span> <span m="3953070">I'm</span> <span m="3953140">going</span> <span m="3953230">to</span> <span m="3953270">be</span> <span m="3953390">fair</span> <span m="3953670">and let</span> <span m="3953830">you</span> <span m="3954050">pick</span> <span m="3954710">first</span> <span m="3955270">which</span> <span m="3955350">one</span> <span m="3955500">you</span> <span m="3955600">like.</span> </p>
<p><span m="3956620">Now,</span> <span m="3956810">as</span> <span m="3957860">you</span> <span m="3957990">look</span> <span m="3958230">at</span> <span m="3958320">this,</span> <span m="3960300">you know,</span> <span m="3960530">you</span> <span m="3960670">guys</span> <span m="3960910">can</span> <span m="3961030">help</span> <span m="3961220">him</span> <span m="3961370">out</span> <span m="3961560">here</span> <span m="3961700">if</span> <span m="3961750">you</span> <span m="3961860">can</span> <span m="3961980">figure</span> <span m="3962250">out</span> <span m="3962430">which</span> <span m="3962600">one's</span> <span m="3962830">best.</span> <span m="3964140">This</span> <span m="3964330">has</span> <span m="3964460">got</span> <span m="3964640">a</span> <span m="3964690">couple</span> <span m="3964960">good</span> <span m="3965180">sized</span> <span m="3965490">numbers.</span> <span m="3966540">You know,</span> <span m="3967340">two</span> <span m="3967560">of</span> <span m="3967620">these</span> <span m="3967890">beat</span> <span m="3968340">these.</span> <span m="3968610">This has</span> <span m="3968800">the</span> <span m="3968890">biggest</span> <span m="3969320">number,</span> <span m="3969920">but it</span> <span m="3970240">also has the</span> <span m="3970500">smallest.</span> <span m="3971600">This</span> <span m="3971790">has</span> <span m="3971950">two</span> <span m="3972920">small</span> <span m="3973370">guys</span> <span m="3973770">in</span> <span m="3973890">it.</span> <span m="3974960">You</span> <span m="3975400">now</span> <span m="3975570">and</span> <span m="3975690">you</span> <span m="3975770">want</span> <span m="3975940">to</span> <span m="3976000">pick</span> <span m="3976210">the</span> <span m="3976310">one</span> <span m="3976790">that's</span> <span m="3977010">likely</span> <span m="3977310">to--</span> </p>
<p><span m="3977620">AUDIENCE: They</span> <span m="3977970">all add up to the same thing.</span> </p>
<p><span m="3979260">PROFESSOR: They</span> <span m="3979410">do</span> <span m="3979720">all</span> <span m="3979900">add up</span> <span m="3980110">to the</span> <span m="3980180">same</span> <span m="3980460">thing,</span> <span m="3980720">that's</span> <span m="3980920">true.</span> <span m="3982060">Yeah.</span> <span m="3985070">Which</span> <span m="3985260">one</span> <span m="3985430">do you</span> <span m="3985560">want?</span> <span m="3998606">Which one do you want here</span> <span m="3999578">Will?</span> <span m="4002666">What's</span> <span m="4003160">your best one there?</span> <span m="4007610">A</span> <span m="4007905">does look</span> <span m="4008200">pretty</span> <span m="4008590">good.</span> <span m="4009418">You've got two</span> <span m="4009896">solid numbers there.</span> <span m="4012286">It does.</span> <span m="4013242">A looks like it</span> <span m="4013720">beats up on</span> <span m="4014200">C</span> <span m="4014410">pretty</span> <span m="4014560">good.</span> </p>
<p><span m="4017560">AUDIENCE: I don't</span> <span m="4018060">know.</span> <span m="4019060">I think</span> <span m="4019560">I'm going to lose a dollar.</span> </p>
<p><span m="4020554">PROFESSOR: That's probably true.</span> </p>
<p><span m="4022045">[LAUGHTER]</span> </p>
<p></p>
<p><span m="4024033">[AUDIENCE CHATTER]</span> </p>
<p></p>
<p><span m="4034967">I don't</span> <span m="4035464">know.</span> <span m="4035961">A looks pretty good to us.</span> <span m="4036480">Which one</span> <span m="4036620">do you like</span> <span m="4036850">here?</span> <span m="4037773">We'll try</span> <span m="4038246">them out.</span> <span m="4039192">We'll</span> <span m="4039665">play</span> <span m="4040138">more games here.</span> <span m="4041560">A.</span> <span m="4042250">All right,</span> <span m="4042500">Will</span> <span m="4042765">likes</span> <span m="4043030">A.</span> <span m="4047712">All right.</span> <span m="4049134">And I'm</span> <span m="4049608">going to</span> <span m="4050082">pick--</span> <span m="4051990">I'm</span> <span m="4052170">going</span> <span m="4052260">to</span> <span m="4052300">go</span> <span m="4052450">with</span> <span m="4052940">C.</span> <span m="4053200">I'm</span> <span m="4053280">going</span> <span m="4053370">to</span> <span m="4053410">give</span> <span m="4053540">you</span> <span m="4053640">a</span> <span m="4053690">fighting</span> <span m="4054010">chance,</span> <span m="4054420">right?</span> <span m="4054710">Because</span> <span m="4054780">we</span> <span m="4054890">figured</span> <span m="4055200">A beat</span> <span m="4055515">C</span> <span m="4055830">for</span> <span m="4055960">sure,</span> <span m="4056250">right?</span> <span m="4057190">Because</span> <span m="4057350">you've</span> <span m="4057420">got</span> <span m="4057550">a</span> <span m="4057590">couple</span> <span m="4057890">of</span> <span m="4058030">6</span> <span m="4058260">and</span> <span m="4058360">7</span> <span m="4058620">beat</span> <span m="4058810">up</span> <span m="4058930">3</span> <span m="4059120">and</span> <span m="4059230">4</span> <span m="4059450">pretty</span> <span m="4059660">good.</span> <span m="4061050">Yeah,</span> <span m="4061520">I'm</span> <span m="4061610">going</span> <span m="4061710">to</span> <span m="4061770">pick</span> <span m="4061990">C.</span> <span m="4064710">All right.</span> </p>
<p><span m="4064830">And now</span> <span m="4065030">we're</span> <span m="4065140">going</span> <span m="4065260">to</span> <span m="4065350">do--</span> <span m="4065900">figure</span> <span m="4066150">out</span> <span m="4066250">which</span> <span m="4066430">one's</span> <span m="4066640">more</span> <span m="4066820">likely</span> <span m="4067100">to</span> <span m="4067190">win.</span> <span m="4069050">And</span> <span m="4069160">we're</span> <span m="4069260">going</span> <span m="4069350">to</span> <span m="4069580">make the</span> <span m="4069670">tree</span> <span m="4069880">diagram.</span> <span m="4071570">All</span> <span m="4071700">right?</span> <span m="4071910">So</span> <span m="4072650">here's</span> <span m="4072940">A.</span> <span m="4075500">You've</span> <span m="4075740">got</span> <span m="4076180">2,</span> <span m="4076630">6,</span> <span m="4077130">and</span> <span m="4077260">7</span> <span m="4077650">are</span> <span m="4077700">your</span> <span m="4077820">choices.</span> <span m="4078290">And</span> <span m="4078380">they're</span> <span m="4078540">each</span> <span m="4078810">1/3.</span> <span m="4081930">And</span> <span m="4082090">then,</span> <span m="4082270">for</span> <span m="4082380">mine,</span> <span m="4082850">I've</span> <span m="4082990">got</span> <span m="4083600">3,</span> <span m="4084160">4,</span> <span m="4084360">and</span> <span m="4084820">8.</span> <span m="4090720">3,</span> <span m="4091100">4,</span> <span m="4091465">and</span> <span m="4091830">8.</span> <span m="4093590">3,</span> <span m="4093910">4,</span> <span m="4094230">and</span> <span m="4094340">8.</span> <span m="4095130">And</span> <span m="4095400">they're</span> <span m="4095600">all</span> <span m="4095790">1/3</span> <span m="4096210">chance.</span> <span m="4099925">All</span> <span m="4100380">right,</span> <span m="4100670">so</span> <span m="4100760">I</span> <span m="4100790">make</span> <span m="4101010">my</span> <span m="4101160">tree</span> <span m="4101760">here.</span> <span m="4103430">I got</span> <span m="4103870">squished at</span> <span m="4104350">the</span> <span m="4104450">bottom,</span> <span m="4105670">but</span> <span m="4105920">they're</span> <span m="4106060">all</span> <span m="4106260">1/3.</span> <span m="4108495">All</span> <span m="4108990">right,</span> <span m="4109189">now</span> <span m="4109370">we're</span> <span m="4109470">going</span> <span m="4109560">to</span> <span m="4109609">see</span> <span m="4109870">who</span> <span m="4110029">wins,</span> <span m="4110939">C or</span> <span m="4111379">A.</span> <span m="4112720">There's</span> <span m="4112939">nine</span> <span m="4113340">outcomes.</span> <span m="4114640">Who</span> <span m="4114800">wins</span> <span m="4115060">this</span> <span m="4115279">outcome?</span> <span m="4118034">Oh,</span> <span m="4118490">I</span> <span m="4118500">do.</span> <span m="4118740">Yeah,</span> <span m="4118930">I</span> <span m="4119010">get</span> <span m="4119160">all</span> <span m="4119310">these.</span> <span m="4120584">That's good.</span> <span m="4121380">You're</span> <span m="4121529">in</span> <span m="4121590">trouble</span> <span m="4121920">already.</span> </p>
<p><span m="4122600">What</span> <span m="4122790">happens</span> <span m="4123130">here,</span> <span m="4123330">though?</span> <span m="4123970">Who</span> <span m="4124090">wins</span> <span m="4124270">that?</span> <span m="4128279">Your</span> <span m="4128410">6</span> <span m="4128770">beats</span> <span m="4129130">3,</span> <span m="4129390">6 beats</span> <span m="4129890">4.</span> <span m="4130010">You win</span> <span m="4130160">those.</span> <span m="4131240">I</span> <span m="4131319">got</span> <span m="4131550">you</span> <span m="4131649">here.</span> <span m="4133609">Then</span> <span m="4133830">you</span> <span m="4134130">win</span> <span m="4134609">7</span> <span m="4135062">beats</span> <span m="4135515">3.</span> <span m="4136210">A</span> <span m="4136490">wins.</span> <span m="4137350">7 beats</span> <span m="4137660">4,</span> <span m="4138580">A</span> <span m="4138760">wins.</span> <span m="4139149">It's</span> <span m="4139300">tied,</span> <span m="4139800">but,</span> <span m="4140060">oh,</span> <span m="4140439">my</span> <span m="4140620">8</span> <span m="4140899">beat your</span> <span m="4141200">7.</span> <span m="4143950">All</span> <span m="4144260">right.</span> <span m="4144609">So</span> <span m="4145779">let's</span> <span m="4146050">see</span> <span m="4146279">here.</span> <span m="4147939">There's</span> <span m="4148260">one,</span> <span m="4148649">two,</span> <span m="4148910">three,</span> <span m="4149380">four,</span> <span m="4149990">five</span> <span m="4150430">for</span> <span m="4150550">me,</span> <span m="4151029">each</span> <span m="4151319">at</span> <span m="4151470">1/9.</span> <span m="4154000">All</span> <span m="4154120">right,</span> <span m="4154420">so</span> <span m="4156960">I'm</span> <span m="4157189">going</span> <span m="4157310">to</span> <span m="4157390">win.</span> <span m="4158583">All</span> <span m="4159010">right?</span> </p>
<p><span m="4160180">Now,</span> <span m="4160470">in</span> <span m="4160600">this</span> <span m="4160800">case,</span> <span m="4162010">the</span> <span m="4162359">sample</span> <span m="4162710">space</span> <span m="4163010">is</span> <span m="4163090">uniform.</span> <span m="4164510">Every</span> <span m="4164979">probability</span> <span m="4165689">is</span> <span m="4165800">the</span> <span m="4165890">same</span> <span m="4166240">for the</span> <span m="4166460">outcomes.</span> <span m="4167140">They're all</span> <span m="4167670">1/9.</span> <span m="4168950">1/3</span> <span m="4169170">times</span> <span m="4169359">1/3.</span> <span m="4169700">1/3</span> <span m="4169960">times</span> <span m="4170200">1/3.</span> <span m="4171390">All</span> <span m="4171500">right?</span> <span m="4171740">So</span> <span m="4171939">that's</span> <span m="4172870">the</span> <span m="4172939">case</span> <span m="4173319">where</span> <span m="4173450">we</span> <span m="4173540">say</span> <span m="4173710">the</span> <span m="4173830">sample</span> <span m="4174210">space</span> <span m="4174590">is</span> <span m="4174700">uniform.</span> <span m="4176090">Don't</span> <span m="4176510">go anywhere.</span> <span m="4176960">We're</span> <span m="4177090">going</span> <span m="4177189">to</span> <span m="4178130">do</span> <span m="4178260">some</span> <span m="4178420">more</span> <span m="4178700">here.</span> <span m="4182180">The</span> <span m="4182270">sample</span> <span m="4182689">space</span> <span m="4183260">is</span> <span m="4184260">uniform</span> <span m="4191140">if</span> <span m="4191470">every</span> <span m="4192310">sample</span> <span m="4192729">point</span> <span m="4193770">has</span> <span m="4194050">the</span> <span m="4194140">same</span> <span m="4194420">probability.</span> <span m="4201880">In</span> <span m="4202010">which</span> <span m="4202230">case,</span> <span m="4203580">the</span> <span m="4203650">probability</span> <span m="4204340">of</span> <span m="4204410">the</span> <span m="4204500">sample</span> <span m="4204910">points</span> <span m="4205350">would</span> <span m="4205500">be</span> <span m="4205670">1</span> <span m="4205990">in</span> <span m="4206260">the</span> <span m="4206350">cardinality</span> <span m="4207060">of</span> <span m="4207150">the</span> <span m="4207250">sample</span> <span m="4207610">space.</span> <span m="4208760">In</span> <span m="4209150">this</span> <span m="4209270">case,</span> <span m="4209480">there's</span> <span m="4209620">nine</span> <span m="4209870">points.</span> <span m="4210320">They each</span> <span m="4210550">have</span> <span m="4210690">probability</span> <span m="4211720">1/9.</span> <span m="4213270">And</span> <span m="4213450">in</span> <span m="4213510">this</span> <span m="4213710">case,</span> <span m="4214510">C</span> <span m="4214855">beats</span> <span m="4215200">A</span> <span m="4218480">with</span> <span m="4218680">probability</span> <span m="4219930">of</span> <span m="4220730">5/9.</span> <span m="4223970">That</span> <span m="4224190">means</span> <span m="4224740">I</span> <span m="4225160">win</span> <span m="4225380">the</span> <span m="4225480">dollar.</span> <span m="4227070">All</span> <span m="4227140">right.</span> </p>
<p><span m="4229058">So</span> <span m="4229950">do you have</span> <span m="4230140">some more</span> <span m="4230290">money</span> <span m="4231462">in there?</span> <span m="4232274">In</span> <span m="4232680">that</span> <span m="4232930">wallet?</span> <span m="4233300">I'll</span> <span m="4233360">make</span> <span m="4233540">change</span> <span m="4233820">for</span> <span m="4234030">you.</span> <span m="4235170">So,</span> <span m="4235490">what</span> <span m="4235650">we're</span> <span m="4235770">going</span> <span m="4235870">to</span> <span m="4235930">do</span> <span m="4236110">is</span> <span m="4236220">play</span> <span m="4236520">again.</span> <span m="4238220">And</span> <span m="4238490">you've</span> <span m="4238630">learned</span> <span m="4238950">something</span> <span m="4239330">now,</span> <span m="4239670">that</span> <span m="4239960">C beats A.</span> <span m="4240910">So</span> <span m="4241730">we</span> <span m="4241900">play</span> <span m="4242140">again,</span> <span m="4242350">you're</span> <span m="4242440">probably</span> <span m="4242750">not</span> <span m="4242920">going</span> <span m="4243030">to</span> <span m="4243090">want</span> <span m="4243230">to</span> <span m="4243300">pick</span> <span m="4244160">A.</span> <span m="4245810">In fact,</span> <span m="4246070">C</span> <span m="4246330">is</span> <span m="4246390">surprisingly</span> <span m="4247100">good</span> <span m="4247380">there.</span> <span m="4249300">Now,</span> <span m="4249500">I'll let</span> <span m="4249720">you</span> <span m="4249800">wager</span> <span m="4250130">more,</span> <span m="4250390">if</span> <span m="4250450">you</span> <span m="4250550">want,</span> <span m="4250740">to</span> <span m="4250820">win</span> <span m="4250940">your</span> <span m="4251020">dollar</span> <span m="4251300">back.</span> <span m="4251410">Do you want to</span> <span m="4251620">do</span> <span m="4251690">double</span> <span m="4251980">or</span> <span m="4252010">nothing?</span> </p>
<p><span m="4253845">[LAUGHTER]</span> </p>
<p></p>
<p><span m="4259335">AUDIENCE: So</span> <span m="4259590">you</span> <span m="4261218">always</span> <span m="4261644">get to pick second, though?</span> </p>
<p><span m="4262500">PROFESSOR: Yeah.</span> <span m="4262850">I've got to</span> <span m="4263220">be fair here.</span> <span m="4264102">I'm going to let you</span> <span m="4264543">pick first.</span> </p>
<p><span m="4265425">[LAUGHTER]</span> </p>
<p><span m="4267470">All right,</span> <span m="4267910">you got a</span> <span m="4268350">couple</span> <span m="4269680">bucks?</span> <span m="4269890">We'll do</span> <span m="4270380">double</span> <span m="4270640">or nothing.</span> <span m="4270990">What</span> <span m="4271100">do you</span> <span m="4271240">say? I'll</span> <span m="4271400">make</span> <span m="4271580">change</span> <span m="4271870">for</span> <span m="4272070">you.</span> <span m="4273270">There you go.</span> <span m="4274690">Here's</span> <span m="4274910">$5.</span> <span m="4275565">I'll change your</span> <span m="4275840">$5.</span> <span m="4277220">And</span> <span m="4277480">then</span> <span m="4277740">we'll</span> <span m="4277940">bet $2.</span> <span m="4278270">How's</span> <span m="4278600">that?</span> <span m="4279390">I've</span> <span m="4279890">actually</span> <span m="4280140">got</span> <span m="4280250">a</span> <span m="4280320">$2</span> <span m="4280720">bill</span> <span m="4280950">here.</span> <span m="4283166">There we go.</span> <span m="4283780">You've</span> <span m="4283960">got $2.</span> <span m="4284440">There</span> <span m="4284855">we go.</span> <span m="4285640">All right,</span> <span m="4286730">double</span> <span m="4287090">or</span> <span m="4287130">nothing.</span> <span m="4288560">Will,</span> <span m="4289020">what do you</span> <span m="4289170">like?</span> <span m="4290030">You're</span> <span m="4290210">obviously</span> <span m="4290660">not</span> <span m="4290860">going</span> <span m="4290960">to</span> <span m="4291020">pick</span> <span m="4291340">A.</span> <span m="4291650">That</span> <span m="4292010">just got</span> <span m="4292160">beat</span> <span m="4292360">by C.</span> <span m="4292570">What</span> <span m="4292740">do you</span> <span m="4292990">like,</span> <span m="4293250">B</span> <span m="4293500">or</span> <span m="4293750">C?</span> </p>
<p><span m="4302724">Well,</span> <span m="4303223">one</span> <span m="4303722">of them's got to</span> <span m="4304221">be best.</span> <span m="4305219">One of them's</span> <span m="4305718">got to be best here.</span> <span m="4306217">You've just got to figure out which</span> <span m="4306716">one it is.</span> </p>
<p><span m="4308712">AUDIENCE: I feel</span> <span m="4309211">like</span> <span m="4309710">everyone's</span> <span m="4310209">going</span> <span m="4310375">to</span> <span m="4310541">have</span> <span m="4310708">[INAUDIBLE].</span> <span m="4311970">If I</span> <span m="4312420">choose C,</span> <span m="4312870">let me</span> <span m="4313250">see.</span> <span m="4314500">That</span> <span m="4314970">would</span> <span m="4315440">be</span> <span m="4316380">[INAUDIBLE]</span> <span m="4318070">so I guess.</span> </p>
<p><span m="4318610">PROFESSOR: I</span> <span m="4318720">won't</span> <span m="4318950">pick</span> <span m="4319390">A,</span> <span m="4319884">that's for sure.</span> </p>
<p><span m="4320872">AUDIENCE: If you pick</span> <span m="4321366">B,</span> <span m="4321860">I get one from 4,</span> <span m="4322354">I get two from the</span> <span m="4322848">8, which</span> <span m="4323342">is four.</span> <span m="4323836">And you get the</span> <span m="4324330">other five.</span> <span m="4325318">If I pick</span> <span m="4325812">C and you pick</span> <span m="4326306">A,</span> <span m="4326800">that</span> <span m="4327294">means I get</span> <span m="4327788">one</span> <span m="4328282">from the</span> <span m="4328776">3,</span> <span m="4329270">I</span> <span m="4329764">get</span> <span m="4330258">one from</span> <span m="4330752">the 4.</span> <span m="4331246">If I get two from the</span> <span m="4331740">8,</span> <span m="4332234">you still get 5.</span> </p>
<p><span m="4334704">PROFESSOR: That's</span> <span m="4335198">good.</span> <span m="4336186">You've</span> <span m="4336680">figured out</span> <span m="4337174">you're</span> <span m="4337668">screwed</span> <span m="4338162">here.</span> <span m="4339644">That's very good.</span> <span m="4340650">All right.</span> <span m="4341980">So,</span> <span m="4342460">yeah,</span> <span m="4342690">so</span> <span m="4342790">he's</span> <span m="4342970">sort</span> <span m="4343260">of--</span> <span m="4343370">he's</span> <span m="4343510">doing</span> <span m="4343750">great.</span> <span m="4344120">He's</span> <span m="4344280">figured</span> <span m="4344550">out,</span> <span m="4344650">because</span> <span m="4344790">he</span> <span m="4344870">went</span> <span m="4345170">through</span> <span m="4345260">this</span> <span m="4345610">sort</span> <span m="4345810">of</span> <span m="4345880">like</span> <span m="4346050">this</span> <span m="4346220">method,</span> <span m="4346640">what</span> <span m="4346780">all</span> <span m="4347040">the</span> <span m="4347130">possible</span> <span m="4347590">outcomes</span> <span m="4347940">were.</span> <span m="4348800">And</span> <span m="4348940">he's</span> <span m="4349100">discovered</span> <span m="4349680">that,</span> <span m="4349890">no</span> <span m="4349980">matter</span> <span m="4350230">what</span> <span m="4350450">he</span> <span m="4350660">picks,</span> <span m="4351680">I'm</span> <span m="4351920">going</span> <span m="4352030">to</span> <span m="4352090">beat</span> <span m="4352280">him.</span> <span m="4353830">So</span> <span m="4353880">let's</span> <span m="4354120">see</span> <span m="4354260">why</span> <span m="4354470">that's</span> <span m="4354700">the</span> <span m="4354790">case.</span> <span m="4355340">So</span> <span m="4356030">let's</span> <span m="4356230">see.</span> <span m="4356440">So</span> <span m="4356990">because</span> <span m="4357290">you</span> <span m="4357390">figured</span> <span m="4357750">it out,</span> <span m="4357870">I</span> <span m="4357940">won't</span> <span m="4358360">take</span> <span m="4358560">your</span> <span m="4358640">money</span> <span m="4359130">this</span> <span m="4359330">time.</span> <span m="4359990">But</span> <span m="4360310">we'll</span> <span m="4360520">do</span> <span m="4360650">some</span> <span m="4360820">more</span> <span m="4360970">in</span> <span m="4361030">a</span> <span m="4361080">minute.</span> </p>
<p><span m="4361770">So</span> <span m="4361810">say</span> <span m="4362130">Will</span> <span m="4362460">picked--</span> <span m="4364220">say he</span> <span m="4364640">picked</span> <span m="4364870">C.</span> <span m="4366900">And</span> <span m="4367180">say</span> <span m="4367420">that</span> <span m="4367560">then</span> <span m="4367730">I</span> <span m="4367920">will</span> <span m="4368120">pick</span> <span m="4370010">B.</span> <span m="4371640">All</span> <span m="4371720">right,</span> <span m="4372000">so</span> <span m="4372150">on</span> <span m="4372450">C,</span> <span m="4372750">there's</span> <span m="4373030">three</span> <span m="4373270">choices.</span> <span m="4375220">3.</span> <span m="4375830">4,</span> <span m="4376290">and</span> <span m="4376410">8.</span> <span m="4377480">And</span> <span m="4377610">then,</span> <span m="4377770">on</span> <span m="4377880">my</span> <span m="4378290">choice</span> <span m="4378680">of</span> <span m="4378790">B,</span> <span m="4379340">there's</span> <span m="4379827">1,</span> <span m="4380314">5,</span> <span m="4380801">and</span> <span m="4381288">9.</span> <span m="4386160">All</span> <span m="4386360">right,</span> <span m="4386630">so</span> <span m="4386720">I</span> <span m="4386770">get</span> <span m="4386920">1,</span> <span m="4387150">5,</span> <span m="4387430">and</span> <span m="4387520">9.</span> <span m="4389120">And</span> <span m="4389290">now</span> <span m="4389500">we</span> <span m="4389610">can</span> <span m="4389740">see</span> <span m="4390080">who</span> <span m="4390220">wins</span> <span m="4390650">here.</span> <span m="4392070">Well,</span> <span m="4392300">C</span> <span m="4392740">wins</span> <span m="4393190">there,</span> <span m="4394850">loses</span> <span m="4395430">here,</span> <span m="4396180">loses</span> <span m="4396590">there.</span> <span m="4397810">C</span> <span m="4398080">wins</span> <span m="4398500">here,</span> <span m="4399970">and</span> <span m="4400300">B</span> <span m="4400760">wins</span> <span m="4401020">the</span> <span m="4401110">next</span> <span m="4401415">two.</span> <span m="4402370">And</span> <span m="4402570">C</span> <span m="4402810">wins</span> <span m="4403100">these.</span> <span m="4404000">And</span> <span m="4404260">B</span> <span m="4404520">wins</span> <span m="4404870">the</span> <span m="4404960">tiebreaker.</span> <span m="4406650">So</span> <span m="4407820">B</span> <span m="4408230">beats</span> <span m="4408470">C</span> <span m="4408950">with</span> <span m="4409020">probability</span> <span m="4409800">5/9.</span> </p>
<p><span m="4418440">All</span> <span m="4418600">right.</span> <span m="4418920">So</span> <span m="4419130">you</span> <span m="4419260">don't</span> <span m="4419540">want</span> <span m="4419720">to</span> <span m="4419780">pick</span> <span m="4420490">A,</span> <span m="4420850">you</span> <span m="4420930">don't</span> <span m="4421130">want</span> <span m="4421270">to</span> <span m="4421320">pick</span> <span m="4421550">C,</span> <span m="4423650">you</span> <span m="4423750">want</span> <span m="4423840">to</span> <span m="4423890">try a</span> <span m="4424160">bet with</span> <span m="4424627">B?</span> <span m="4428363">I</span> <span m="4428830">mean,</span> <span m="4429060">look,</span> <span m="4429270">you've</span> <span m="4429380">got</span> <span m="4430110">B beats</span> <span m="4430466">C.</span> <span m="4431620">We</span> <span m="4431710">just</span> <span m="4431900">showed</span> <span m="4432200">that.</span> <span m="4432840">C beats</span> <span m="4433110">A.</span> <span m="4434610">I</span> <span m="4434680">mean,</span> <span m="4434800">that's</span> <span m="4435010">a</span> <span m="4435050">sure</span> <span m="4435320">bet</span> <span m="4435510">if</span> <span m="4435640">I</span> <span m="4435700">ever</span> <span m="4435910">saw</span> <span m="4436160">one.</span> </p>
<p><span m="4437847">[LAUGHTER]</span> </p>
<p><span m="4439284">PROFESSOR: What do you think?</span> </p>
<p><span m="4441200">AUDIENCE: So</span> <span m="4441520">if I</span> <span m="4441840">pick</span> <span m="4442332">B</span> <span m="4442824">and you</span> <span m="4443316">pick</span> <span m="4443808">C,</span> <span m="4444792">that</span> <span m="4445284">means</span> <span m="4445776">[INAUDIBLE]</span> </p>
<p><span m="4449220">[AUDIENCE CHATTER]</span> </p>
<p><span m="4452172">But if</span> <span m="4452664">he</span> <span m="4453156">picks</span> <span m="4454140">A,</span> <span m="4456108">then</span> <span m="4456600">I</span> <span m="4457092">get</span> <span m="4457584">[INAUDIBLE]</span> <span m="4461520">Yeah,</span> <span m="4462012">I'd still get</span> <span m="4462504">4,</span> <span m="4462996">so--</span> </p>
<p><span m="4464010">PROFESSOR: So</span> <span m="4464220">if you</span> <span m="4464430">pick</span> <span m="4464650">B,</span> <span m="4464840">what am</span> <span m="4464970">I going</span> <span m="4465180">to</span> <span m="4465340">do?</span> </p>
<p><span m="4466752">AUDIENCE: Oh,</span> <span m="4467228">you're going to pick</span> <span m="4468180">A.</span> </p>
<p><span m="4468656">PROFESSOR: I'm going</span> <span m="4469132">to pick A.</span> <span m="4470084">Let's see how that works out.</span> <span m="4471040">All</span> <span m="4471270">right,</span> <span m="4471500">so</span> <span m="4472590">B</span> <span m="4473340">has</span> <span m="4474210">1,</span> <span m="4474530">5,</span> <span m="4475000">and</span> <span m="4475462">9.</span> <span m="4476850">A</span> <span m="4477542">is,</span> <span m="4479270">yeah,</span> <span m="4479650">2,</span> <span m="4479860">6,</span> <span m="4480240">and</span> <span m="4480360">7.</span> <span m="4486780">OK,</span> <span m="4487270">so</span> <span m="4488470">you're</span> <span m="4488750">going</span> <span m="4488830">to</span> <span m="4488900">lose</span> <span m="4489200">all</span> <span m="4489400">three</span> <span m="4489590">there.</span> <span m="4490670">A</span> <span m="4491060">wins</span> <span m="4491370">those.</span> <span m="4493320">A</span> <span m="4493560">wins--</span> <span m="4495670">no,</span> <span m="4495890">you</span> <span m="4496070">win.</span> <span m="4496950">B</span> <span m="4497170">wins</span> <span m="4498400">there,</span> <span m="4499400">but</span> <span m="4499500">A</span> <span m="4499680">wins the</span> <span m="4499920">next</span> <span m="4500240">couple.</span> <span m="4501190">And</span> <span m="4501310">then,</span> <span m="4501440">you</span> <span m="4501550">win</span> <span m="4501790">the</span> <span m="4501890">last three.</span> </p>
<p><span m="4505500">So</span> <span m="4509220">A</span> <span m="4509520">wins,</span> <span m="4509950">again,</span> <span m="4510400">5/9</span> <span m="4511160">of</span> <span m="4511210">the</span> <span m="4511320">time.</span> <span m="4513090">So</span> <span m="4513270">how</span> <span m="4513800">is</span> <span m="4514000">this</span> <span m="4514250">possible?</span> <span m="4515880">We've</span> <span m="4516060">got</span> <span m="4519050">A</span> <span m="4519440">beats</span> <span m="4520284">B,</span> <span m="4521130">beats</span> <span m="4521190">C,</span> <span m="4522660">beats</span> <span m="4522930">A.</span> <span m="4523682">All</span> <span m="4524060">5/9.</span> <span m="4526040">How</span> <span m="4526470">is</span> <span m="4526830">that possible?</span> <span m="4527885">Did we</span> <span m="4528340">make a</span> <span m="4528795">mistake?</span> <span m="4532440">Well,</span> <span m="4533610">why</span> <span m="4533910">isn't</span> <span m="4534110">it</span> <span m="4534310">possible?</span> <span m="4535610">Whoever</span> <span m="4536040">said</span> <span m="4536340">probabilities</span> <span m="4537160">were</span> <span m="4537360">transitive?</span> <span m="4540050">Right?</span> <span m="4541002">I</span> <span m="4541360">mean,</span> <span m="4541470">you'd</span> <span m="4541590">sort</span> <span m="4541840">of</span> <span m="4541940">think</span> <span m="4542210">there's</span> <span m="4542370">a</span> <span m="4542430">best</span> <span m="4542890">of</span> <span m="4542950">the</span> <span m="4543050">three</span> <span m="4543350">die.</span> <span m="4544260">But</span> <span m="4544410">for</span> <span m="4544540">any</span> <span m="4544790">one</span> <span m="4545110">that you</span> <span m="4545280">pick,</span> <span m="4545690">that</span> <span m="4545730">he</span> <span m="4545900">picks,</span> <span m="4546220">I</span> <span m="4546340">can</span> <span m="4546520">find</span> <span m="4546710">one</span> <span m="4546870">that's</span> <span m="4547040">better.</span> <span m="4547290">And</span> <span m="4547360">it</span> <span m="4547420">goes</span> <span m="4547650">in</span> <span m="4547740">a</span> <span m="4547790">cycle.</span> <span m="4549840">All</span> <span m="4549910">right?</span> </p>
<p><span m="4550130">So</span> <span m="4550730">you'd</span> <span m="4550940">think</span> <span m="4551290">picking</span> <span m="4551600">first,</span> <span m="4552070">you've</span> <span m="4552150">got</span> <span m="4552300">a</span> <span m="4552340">shot</span> <span m="4552820">to</span> <span m="4552890">pick</span> <span m="4553090">the</span> <span m="4553170">best</span> <span m="4553490">one.</span> <span m="4553780">But picking</span> <span m="4554050">second</span> <span m="4554890">always</span> <span m="4555350">wins</span> <span m="4555730">this</span> <span m="4555930">game,</span> <span m="4556890">if</span> <span m="4556990">you</span> <span m="4557080">pick</span> <span m="4557250">the</span> <span m="4557340">right</span> <span m="4557540">one.</span> <span m="4559510">All right.</span> <span m="4559910">You</span> <span m="4560390">know,</span> <span m="4560550">since</span> <span m="4560770">you</span> <span m="4560880">did</span> <span m="4561080">good</span> <span m="4561270">and</span> <span m="4561380">figured</span> <span m="4561600">that</span> <span m="4561820">out,</span> <span m="4562280">I'm</span> <span m="4562430">going</span> <span m="4562520">to</span> <span m="4562570">give</span> <span m="4562760">you</span> <span m="4562880">a</span> <span m="4562940">special</span> <span m="4563400">opportunity</span> <span m="4564000">here.</span> <span m="4565920">All</span> <span m="4566030">right?</span> </p>
<p><span m="4566350">Now,</span> <span m="4566970">I'm</span> <span m="4567230">going</span> <span m="4567330">to</span> <span m="4567410">let</span> <span m="4569160">you</span> <span m="4569340">go</span> <span m="4569510">second.</span> <span m="4571540">All right?</span> <span m="4573030">So</span> <span m="4573270">I'm</span> <span m="4573540">going</span> <span m="4573640">to</span> <span m="4573720">pick</span> <span m="4573960">one,</span> <span m="4575300">and</span> <span m="4575670">I'm going to</span> <span m="4575880">let</span> <span m="4576040">you</span> <span m="4576170">pick</span> <span m="4576410">after</span> <span m="4576720">me.</span> <span m="4577360">And</span> <span m="4577490">then</span> <span m="4577640">we're</span> <span m="4577770">going</span> <span m="4577860">to</span> <span m="4577930">bet.</span> <span m="4579480">There's</span> <span m="4579630">going to</span> <span m="4579840">be a</span> <span m="4579900">little</span> <span m="4580120">twist</span> <span m="4580430">to</span> <span m="4580550">it.</span> <span m="4582900">But</span> <span m="4583040">you</span> <span m="4583150">want</span> <span m="4583410">to</span> <span m="4583480">pick</span> <span m="4583680">second,</span> <span m="4584050">right?</span> <span m="4588291">All right,</span> <span m="4588710">well,</span> <span m="4588890">let me tell you what</span> <span m="4589175">I'm going to</span> <span m="4589460">do.</span> <span m="4590485">Just to make it--</span> <span m="4590830">I want to</span> <span m="4590890">do</span> <span m="4591020">a</span> <span m="4591270">different</span> <span m="4591550">analysis.</span> <span m="4592030">We</span> <span m="4592130">already</span> <span m="4592330">did</span> <span m="4592560">these</span> <span m="4592740">three,</span> <span m="4592970">so what</span> <span m="4593160">I'm</span> <span m="4593310">going</span> <span m="4593400">to</span> <span m="4593460">do</span> <span m="4594120">is an</span> <span m="4594300">analysis</span> <span m="4595070">where</span> <span m="4595260">we</span> <span m="4595670">roll</span> <span m="4596040">them twice.</span> <span m="4597340">Otherwise, it's</span> <span m="4597820">the</span> <span m="4597890">same</span> <span m="4598250">basic</span> <span m="4598650">thing,</span> <span m="4598830">but I</span> <span m="4598930">want</span> <span m="4599130">to</span> <span m="4599200">do an</span> <span m="4599380">analysis</span> <span m="4600100">with more</span> <span m="4600360">outcomes</span> <span m="4600620">here.</span> <span m="4601510">We've</span> <span m="4601680">sort</span> <span m="4601890">of</span> <span m="4602040">beat this</span> <span m="4602390">one to death.</span> </p>
<p><span m="4603310">So</span> <span m="4603540">I'm</span> <span m="4603720">going</span> <span m="4603810">to</span> <span m="4603850">go</span> <span m="4604030">first.</span> <span m="4605210">And</span> <span m="4608210">let's</span> <span m="4608460">see,</span> <span m="4608610">what</span> <span m="4609750">do I</span> <span m="4609810">want</span> <span m="4610000">to</span> <span m="4610150">pick?</span> <span m="4612860">Hm.</span> <span m="4614520">Well,</span> <span m="4614960">what should</span> <span m="4615400">I pick,</span> <span m="4615840">if I'm going to do this?</span> <span m="4616720">I</span> <span m="4616810">guess</span> <span m="4617170">I'll</span> <span m="4617440">pick</span> <span m="4619730">B,</span> <span m="4620930">all</span> <span m="4621060">right?</span> <span m="4621330">So</span> <span m="4621450">I'm</span> <span m="4621640">going</span> <span m="4621720">to</span> <span m="4621810">pick</span> <span m="4622270">B</span> <span m="4622730">here.</span> <span m="4627340">All right.</span> <span m="4627440">How much do you</span> <span m="4627520">want</span> <span m="4627630">to bet</span> <span m="4627670">here?</span> <span m="4627970">I'm going to</span> <span m="4628210">let you</span> <span m="4628310">win</span> <span m="4628460">your</span> <span m="4628560">money</span> <span m="4628800">back.</span> <span m="4629050">I'll let you</span> <span m="4629380">bet</span> <span m="4629590">up</span> <span m="4629690">to</span> <span m="4629810">$10.</span> <span m="4631630">Your</span> <span m="4631810">choice.</span> <span m="4636100">What</span> <span m="4636260">do</span> <span m="4636350">you--</span> <span m="4636500">what do you want to</span> <span m="4636720">bet here?</span> <span m="4636970">We've got</span> <span m="4637240">to</span> <span m="4637280">put</span> <span m="4637430">some</span> <span m="4637540">money</span> <span m="4637710">on</span> <span m="4637780">the</span> <span m="4637870">table,</span> <span m="4638280">make it</span> <span m="4638460">a</span> <span m="4638490">little</span> <span m="4638730">interesting.</span> <span m="4642190">Yeah, you</span> <span m="4642450">pick--</span> </p>
<p><span m="4643337">AUDIENCE: Do you</span> <span m="4643834">roll a different die</span> <span m="4644331">or what?</span> </p>
<p><span m="4645230">PROFESSOR: No, the</span> <span m="4645520">same ones.</span> <span m="4645980">The</span> <span m="4646270">die</span> <span m="4646600">don't</span> <span m="4646870">change.</span> <span m="4647680">I'm</span> <span m="4647880">picking</span> <span m="4648190">B.</span> <span m="4649276">And</span> <span m="4649762">I'm going to</span> <span m="4650248">roll it</span> <span m="4650734">a couple times.</span> <span m="4651220">And I'm</span> <span m="4651380">going</span> <span m="4651470">to</span> <span m="4651550">add</span> <span m="4651770">up</span> <span m="4651900">my</span> <span m="4652060">score.</span> <span m="4652950">And</span> <span m="4653390">then, you</span> <span m="4653470">get to</span> <span m="4653670">pick</span> <span m="4653840">whatever</span> <span m="4654100">die</span> <span m="4654360">you</span> <span m="4654500">want.</span> <span m="4654700">I'm going</span> <span m="4654760">to let</span> <span m="4654870">you pick.</span> <span m="4655670">I</span> <span m="4655770">mean,</span> <span m="4656700">I'm</span> <span m="4656980">going to</span> <span m="4657800">let you</span> <span m="4658560">pick</span> <span m="4658790">the</span> <span m="4658970">one</span> <span m="4659210">that beats</span> <span m="4659490">it.</span> <span m="4659770">You</span> <span m="4659930">roll</span> <span m="4660110">yours</span> <span m="4660390">a</span> <span m="4660430">couple</span> <span m="4660690">times.</span> <span m="4661620">And</span> <span m="4661630">now</span> <span m="4661880">we</span> <span m="4662020">see</span> <span m="4662260">which</span> <span m="4662460">one</span> <span m="4662630">has</span> <span m="4662860">the</span> <span m="4662910">higher</span> <span m="4663310">sum,</span> <span m="4664304">who's</span> <span m="4664801">more likely</span> <span m="4665298">to win.</span> </p>
<p><span m="4666292">AUDIENCE: So it's</span> <span m="4666789">the sum</span> <span m="4667286">total, like you roll a 9 and 9 and</span> <span m="4667783">I roll a 7 and 7.</span> </p>
<p><span m="4668777">PROFESSOR: Yeah,</span> <span m="4669280">then</span> <span m="4669590">I'm going to</span> <span m="4669910">win,</span> <span m="4670060">because I get</span> <span m="4670200">18</span> <span m="4670610">and you</span> <span m="4670870">got</span> <span m="4671090">14.</span> <span m="4672090">If I</span> <span m="4672270">roll</span> <span m="4672595">a</span> <span m="4672920">9 and a</span> <span m="4673210">1,</span> <span m="4673270">I get a</span> <span m="4673410">10.</span> <span m="4674270">You roll</span> <span m="4674680">a 6</span> <span m="4675090">and a 7,</span> <span m="4675345">you beat me</span> <span m="4675600">at 13.</span> <span m="4676884">All right?</span> <span m="4678360">So</span> <span m="4679660">which</span> <span m="4679840">one</span> <span m="4680010">do you</span> <span m="4680090">want</span> <span m="4680210">to</span> <span m="4680640">pick?</span> <span m="4682490">You</span> <span m="4682745">could</span> <span m="4683000">pick</span> <span m="4683375">C.</span> <span m="4684520">But you've got</span> <span m="4684740">to</span> <span m="4684810">put</span> <span m="4685000">the</span> <span m="4685080">money</span> <span m="4685330">on</span> <span m="4685390">the</span> <span m="4685470">table</span> <span m="4685740">here,</span> <span m="4687924">whatever</span> <span m="4688395">you want</span> <span m="4688870">to bet.</span> <span m="4689410">You know,</span> <span m="4689730">I want to</span> <span m="4689830">give you</span> <span m="4690040">a chance</span> <span m="4690340">to win</span> <span m="4690500">your</span> <span m="4690590">money</span> <span m="4690810">back.</span> <span m="4694830">What's</span> <span m="4695030">the</span> <span m="4695110">probability</span> <span m="4695730">he's</span> <span m="4695820">going to lose?</span> </p>
<p><span m="4698032">[LAUGHTER]</span> </p>
<p><span m="4699514">You got a</span> <span m="4700008">T-Card?</span> <span m="4700502">What've you got in</span> <span m="4700996">there?</span> <span m="4701490">Ah,</span> <span m="4701630">I'll take</span> <span m="4701700">the</span> <span m="4702030">[INAUDIBLE]</span> <span m="4702530">card.</span> <span m="4703512">What's that</span> <span m="4704003">worth?</span> <span m="4704985">Ah,</span> <span m="4705476">it's worth a few</span> <span m="4705967">bucks</span> <span m="4706458">here.</span> <span m="4707931">All right,</span> <span m="4709404">so</span> <span m="4711000">I'm</span> <span m="4711110">picking</span> <span m="4711260">B.</span> <span m="4712840">Which</span> <span m="4713020">are you</span> <span m="4713190">picking?</span> <span m="4717110">I've really</span> <span m="4717380">got</span> <span m="4717520">his</span> <span m="4717650">head</span> <span m="4718000">messed up</span> <span m="4718300">now.</span> </p>
<p><span m="4720870">AUDIENCE: There's</span> <span m="4721050">$4 on the</span> <span m="4721420">table.</span> <span m="4721720">Take that</span> <span m="4721900">and</span> <span m="4722120">run.</span> </p>
<p><span m="4724040">PROFESSOR: Yeah, really.</span> <span m="4724690">We</span> <span m="4725030">already</span> <span m="4725415">saw--</span> <span m="4725800">I'll give you</span> <span m="4726020">a</span> <span m="4726060">hint</span> <span m="4726270">here.</span> <span m="4726570">A</span> <span m="4726800">beats B.</span> <span m="4727860">5/9</span> <span m="4728540">of the</span> <span m="4728680">time,</span> <span m="4729010">A</span> <span m="4729190">comes</span> <span m="4729410">up</span> <span m="4729520">higher</span> <span m="4729790">than</span> <span m="4729930">B.</span> <span m="4730860">So</span> <span m="4730970">if</span> <span m="4731040">we</span> <span m="4731160">roll them</span> <span m="4731610">twice,</span> <span m="4733540">your odds</span> <span m="4734030">got even</span> <span m="4734520">better</span> <span m="4735010">that</span> <span m="4735500">A beats</span> <span m="4735990">B.</span> <span m="4738930">All right,</span> <span m="4739420">so Will goes with A.</span> <span m="4740890">Will,</span> <span m="4741380">that</span> <span m="4741870">was a big mistake.</span> <span m="4743850">All</span> <span m="4744100">right.</span> <span m="4745295">How can it be</span> <span m="4745790">a mistake?</span> <span m="4746020">A beats</span> <span m="4746380">B</span> <span m="4747310">5/9</span> <span m="4747620">of</span> <span m="4747930">the</span> <span m="4748060">time.</span> <span m="4749220">Now</span> <span m="4749430">what</span> <span m="4749530">we're</span> <span m="4749550">going</span> <span m="4749630">to</span> <span m="4749700">do</span> <span m="4750530">is</span> <span m="4750690">make</span> <span m="4750930">the</span> <span m="4751000">decision</span> <span m="4751420">tree.</span> <span m="4752220">If</span> <span m="4752510">we</span> <span m="4752620">roll</span> <span m="4752910">B</span> <span m="4753380">twice,</span> <span m="4753790">if we</span> <span m="4753910">roll</span> <span m="4754240">A</span> <span m="4754410">twice.</span> <span m="4755445">All right.</span> <span m="4755850">Let's</span> <span m="4756160">see</span> <span m="4756310">what</span> <span m="4756460">happens.</span> <span m="4757530">This</span> <span m="4757700">is</span> <span m="4757800">a</span> <span m="4757990">nasty</span> <span m="4758380">one</span> <span m="4758550">to</span> <span m="4758620">write</span> <span m="4758850">down.</span> <span m="4759090">In</span> <span m="4759160">fact,</span> <span m="4759360">you'll</span> <span m="4759450">start</span> <span m="4759730">seeing</span> <span m="4760030">this</span> <span m="4760170">when</span> <span m="4760290">you</span> <span m="4760370">do</span> <span m="4760510">these</span> <span m="4760780">tree</span> <span m="4760980">method</span> <span m="4761320">things.</span> <span m="4762100">Sometimes</span> <span m="4762510">the</span> <span m="4762580">trees</span> <span m="4762890">get</span> <span m="4762970">a</span> <span m="4763010">little</span> <span m="4763180">hairy,</span> <span m="4763560">so</span> <span m="4763710">we've</span> <span m="4763800">got</span> <span m="4763950">to</span> <span m="4763990">sort</span> <span m="4764180">of</span> <span m="4764670">approximate</span> <span m="4764960">it</span> <span m="4765250">a</span> <span m="4765300">little</span> <span m="4765460">bit.</span> </p>
<p><span m="4767520">So</span> <span m="4768030">here's</span> <span m="4768560">B,</span> <span m="4769880">right?</span> <span m="4770140">I</span> <span m="4770210">got</span> <span m="4770540">1,</span> <span m="4770820">5,</span> <span m="4771140">and</span> <span m="4771220">9</span> <span m="4771640">for</span> <span m="4771740">my</span> <span m="4771850">first</span> <span m="4772240">roll.</span> <span m="4773950">And</span> <span m="4774140">then</span> <span m="4774280">I</span> <span m="4774360">got</span> <span m="4774610">1,</span> <span m="4774910">5,</span> <span m="4775220">and</span> <span m="4775310">9</span> <span m="4775580">for</span> <span m="4775680">my</span> <span m="4775760">second</span> <span m="4776140">roll.</span> <span m="4782100">And</span> <span m="4782260">I</span> <span m="4782350">add</span> <span m="4782660">these</span> <span m="4782880">up.</span> <span m="4784590">See,</span> <span m="4784980">if you</span> <span m="4785250">read</span> <span m="4785340">the</span> <span m="4785615">notes</span> <span m="4785890">ahead,</span> <span m="4786170">sometimes</span> <span m="4786560">that</span> <span m="4786700">helps.</span> <span m="4787990">6--</span> <span m="4788940">here</span> <span m="4789240">are my</span> <span m="4789440">outcomes.</span> <span m="4789960">These are</span> <span m="4790130">the</span> <span m="4790240">numbers</span> <span m="4790660">I'm</span> <span m="4790680">going</span> <span m="4790790">to</span> <span m="4790840">get.</span> <span m="4794300">These</span> <span m="4794520">are</span> <span m="4794570">my</span> <span m="4794860">nine</span> <span m="4795960">values.</span> </p>
<p><span m="4797000">Your</span> <span m="4797300">nine</span> <span m="4797660">values,</span> <span m="4798300">you</span> <span m="4798440">got</span> <span m="4798720">2,</span> <span m="4798890">6,</span> <span m="4799180">and</span> <span m="4799310">7.</span> <span m="4801360">Then</span> <span m="4801610">each 2, 6, and 7</span> <span m="4802400">again.</span> <span m="4805170">So</span> <span m="4805600">we</span> <span m="4805770">add</span> <span m="4805970">those</span> <span m="4806230">up,</span> <span m="4806420">you</span> <span m="4806510">got</span> <span m="4806800">4, 8, 9, 8,</span> <span m="4808300">12,</span> <span m="4810300">13,</span> <span m="4812242">9,</span> <span m="4813126">13,</span> <span m="4814010">and</span> <span m="4814140">14.</span> <span m="4815165">And</span> <span m="4815800">now,</span> <span m="4816110">we</span> <span m="4816220">have</span> <span m="4816470">each</span> <span m="4816690">one</span> <span m="4816850">of</span> <span m="4816930">these</span> <span m="4817290">against</span> <span m="4817680">each</span> <span m="4817860">one</span> <span m="4818020">of</span> <span m="4818110">these.</span> <span m="4819130">Because</span> <span m="4819270">there's</span> <span m="4819520">81</span> <span m="4820170">outcomes.</span> <span m="4820830">I</span> <span m="4820910">got</span> <span m="4821120">four</span> <span m="4821410">levels</span> <span m="4821710">to</span> <span m="4821800">the</span> <span m="4821890">tree.</span> <span m="4822230">And</span> <span m="4822350">rather</span> <span m="4822620">than</span> <span m="4822760">draw</span> <span m="4823780">four</span> <span m="4824060">levels</span> <span m="4824430">of</span> <span m="4824490">this</span> <span m="4824670">[INAUDIBLE]</span> <span m="4825130">tree</span> <span m="4825370">to</span> <span m="4825410">get</span> <span m="4825550">81</span> <span m="4825820">outcomes,</span> <span m="4826860">you're</span> <span m="4826940">going</span> <span m="4827030">to</span> <span m="4827080">say</span> <span m="4827250">that</span> <span m="4827410">each</span> <span m="4827650">one</span> <span m="4827920">of</span> <span m="4828020">these</span> <span m="4828780">corresponds</span> <span m="4830280">to</span> <span m="4830360">this</span> <span m="4830550">tree,</span> <span m="4832025">all</span> <span m="4832410">right,</span> <span m="4832640">that</span> <span m="4832730">has</span> <span m="4833000">nine</span> <span m="4833320">leaves.</span> <span m="4834330">And</span> <span m="4834460">now</span> <span m="4834650">we</span> <span m="4834740">can</span> <span m="4834850">just</span> <span m="4835050">count</span> <span m="4835400">up</span> <span m="4836100">how</span> <span m="4836320">many</span> <span m="4836580">times</span> <span m="4837480">I</span> <span m="4837630">win.</span> </p>
<p><span m="4840205">Well,</span> <span m="4840630">2</span> <span m="4840860">got</span> <span m="4841140">nothing.</span> <span m="4841630">That's</span> <span m="4842005">a</span> <span m="4842380">0.</span> <span m="4844270">Let's</span> <span m="4844420">see how many</span> <span m="4844510">times</span> <span m="4844680">6</span> <span m="4845030">wins.</span> <span m="4846120">One--</span> <span m="4847650">one</span> <span m="4847860">time.</span> <span m="4850064">10--</span> <span m="4851010">one,</span> <span m="4851280">two,</span> <span m="4851560">three,</span> <span m="4851960">four,</span> <span m="4852870">five.</span> <span m="4855890">All right,</span> <span m="4856360">we've got that.</span> <span m="4857050">6</span> <span m="4857540">wins,</span> <span m="4857840">we</span> <span m="4858070">already</span> <span m="4858160">decided,</span> <span m="4858600">once.</span> <span m="4861260">10,</span> <span m="4861880">we</span> <span m="4862000">decided</span> <span m="4862490">already,</span> <span m="4862910">wins</span> <span m="4863420">five</span> <span m="4863878">times.</span> <span m="4866630">14</span> <span m="4866920">wins</span> <span m="4867120">one,</span> <span m="4867560">two,</span> <span m="4868000">three,</span> <span m="4868440">four,</span> <span m="4868880">five,</span> <span m="4869320">six, seven, eight,</span> <span m="4870200">and</span> <span m="4870620">ties once.</span> <span m="4872290">Tie--</span> <span m="4872760">one.</span> <span m="4875390">10,</span> <span m="4875790">we</span> <span m="4875820">already</span> <span m="4876050">got</span> <span m="4876430">was</span> <span m="4876630">five.</span> <span m="4878550">14,</span> <span m="4879510">we</span> <span m="4879620">already</span> <span m="4879810">got,</span> <span m="4880620">was</span> <span m="4880970">eight</span> <span m="4881390">and a</span> <span m="4881510">tie.</span> <span m="4883260">And</span> <span m="4883770">finally,</span> <span m="4884110">I</span> <span m="4884160">get</span> <span m="4884510">18</span> <span m="4885230">wins</span> <span m="4885890">every</span> <span m="4886210">time.</span> <span m="4888000">Let's</span> <span m="4888190">add</span> <span m="4888350">this</span> <span m="4888540">up.</span> <span m="4888780">I got</span> <span m="4889100">17, 22,</span> <span m="4891210">30,</span> <span m="4892154">35,</span> <span m="4893098">36,</span> <span m="4894042">41,</span> <span m="4894986">42</span> <span m="4895460">times</span> <span m="4896210">plus</span> <span m="4896820">half</span> <span m="4897060">the</span> <span m="4897220">ties</span> <span m="4898570">out</span> <span m="4898900">of</span> <span m="4900360">81.</span> <span m="4902680">How</span> <span m="4902810">much</span> <span m="4903030">money's</span> <span m="4903350">on</span> <span m="4903450">this</span> <span m="4903550">[INAUDIBLE]</span> <span m="4903890">card?</span> <span m="4909224">Ah,</span> <span m="4909690">smart</span> <span m="4910060">guy.</span> </p>
<p><span m="4913919">How did this happen?</span> <span m="4915410">A</span> <span m="4915685">is</span> <span m="4915960">more</span> <span m="4916300">likely</span> <span m="4916890">to</span> <span m="4916980">beat</span> <span m="4917365">B,</span> <span m="4917750">higher</span> <span m="4918020">than</span> <span m="4918180">B.</span> <span m="4918550">But</span> <span m="4918650">you</span> <span m="4918790">roll them</span> <span m="4919230">twice,</span> <span m="4919750">and</span> <span m="4919920">it's</span> <span m="4920160">less</span> <span m="4920570">likely</span> <span m="4920920">to</span> <span m="4920980">be</span> <span m="4921140">as large</span> <span m="4921523">on the</span> <span m="4921906">sum?</span> <span m="4923770">Well,</span> <span m="4923900">why</span> <span m="4924100">not?</span> <span m="4925020">It's</span> <span m="4925360">just</span> <span m="4925540">our</span> <span m="4925670">intuition's</span> <span m="4926320">all</span> <span m="4926500">screwed</span> <span m="4926820">up.</span> <span m="4927760">But</span> <span m="4927900">the</span> <span m="4927990">math</span> <span m="4928320">is</span> <span m="4928430">straightforward.</span> <span m="4930160">In</span> <span m="4930610">fact, an amazing</span> <span m="4931060">thing happens.</span> <span m="4932410">If</span> <span m="4932580">you</span> <span m="4932750">roll</span> <span m="4933040">them</span> <span m="4933220">twice,</span> <span m="4935350">they</span> <span m="4935510">reverse</span> <span m="4936420">completely.</span> <span m="4937710">A</span> <span m="4937920">is</span> <span m="4938080">worse</span> <span m="4938340">than</span> <span m="4938470">B,</span> <span m="4939320">is</span> <span m="4939460">worse</span> <span m="4939750">than</span> <span m="4939870">C,</span> <span m="4940470">is</span> <span m="4940600">worse</span> <span m="4940990">than A</span> <span m="4942590">if</span> <span m="4942710">you</span> <span m="4942830">roll them</span> <span m="4943100">twice.</span> <span m="4944150">If</span> <span m="4944300">you</span> <span m="4944420">roll them</span> <span m="4944730">three</span> <span m="4945070">times,</span> <span m="4945510">something</span> <span m="4945860">else</span> <span m="4946100">happens.</span> </p>
<p><span m="4947570">In</span> <span m="4947670">fact,</span> <span m="4948120">it</span> <span m="4948200">was</span> <span m="4948360">known</span> <span m="4948590">for</span> <span m="4948690">a</span> <span m="4948740">long</span> <span m="4949050">time,</span> <span m="4949540">this</span> <span m="4949730">phenomenon</span> <span m="4950190">happens</span> <span m="4950560">with</span> <span m="4950660">one</span> <span m="4950930">roll.</span> <span m="4951665">And</span> <span m="4951930">when I</span> <span m="4952120">was making</span> <span m="4952420">lecture</span> <span m="4952730">notes</span> <span m="4953310">a</span> <span m="4953390">few</span> <span m="4953570">years</span> <span m="4953820">ago,</span> <span m="4954070">I</span> <span m="4954170">noticed</span> <span m="4954380">they</span> <span m="4954460">reverse</span> <span m="4954900">with</span> <span m="4955050">two</span> <span m="4955250">rolls.</span> <span m="4955970">And</span> <span m="4956060">then</span> <span m="4956160">we</span> <span m="4956240">did a</span> <span m="4956390">class</span> <span m="4956760">project</span> <span m="4957240">to</span> <span m="4957300">discover</span> <span m="4957660">what</span> <span m="4957840">happens</span> <span m="4958230">with</span> <span m="4958400">k</span> <span m="4958650">rolls.</span> <span m="4959830">And</span> <span m="4960040">it</span> <span m="4960150">turns</span> <span m="4960500">out</span> <span m="4960870">that</span> <span m="4960970">you</span> <span m="4961070">can</span> <span m="4961190">get</span> <span m="4961390">every</span> <span m="4961710">possible</span> <span m="4962740">combination.</span> <span m="4964200">You</span> <span m="4964290">know,</span> <span m="4964440">on</span> <span m="4964570">some</span> <span m="4964730">number</span> <span m="4964980">of</span> <span m="4965060">rolls,</span> <span m="4965450">A</span> <span m="4965680">beats</span> <span m="4965940">both</span> <span m="4966200">B</span> <span m="4966440">and</span> <span m="4966670">C</span> <span m="4967050">and</span> <span m="4967160">B</span> <span m="4967350">beats</span> <span m="4967620">C.</span> <span m="4968730">Other</span> <span m="4969360">number</span> <span m="4969580">of</span> <span m="4969660">rolls,</span> <span m="4970230">A</span> <span m="4970400">beats</span> <span m="4970560">them</span> <span m="4970700">both</span> <span m="4970940">and</span> <span m="4971030">C</span> <span m="4971410">beats</span> <span m="4971745">B.</span> </p>
<p><span m="4973010">And</span> <span m="4973120">then,</span> <span m="4973270">motivated</span> <span m="4973810">by</span> <span m="4973950">that,</span> <span m="4974590">some</span> <span m="4974840">researchers</span> <span m="4975600">out</span> <span m="4975750">in</span> <span m="4975830">San</span> <span m="4976080">Diego</span> <span m="4977160">showed</span> <span m="4977630">that</span> <span m="4977760">there's</span> <span m="4978220">arbitrarily</span> <span m="4979330">strange</span> <span m="4980070">dice,</span> <span m="4981940">now,</span> <span m="4982300">not</span> <span m="4982560">with</span> <span m="4982680">just</span> <span m="4982950">three sides,</span> <span m="4983600">but</span> <span m="4983710">with</span> <span m="4983830">more</span> <span m="4984070">sides.</span> <span m="4985830">Such</span> <span m="4986160">that,</span> <span m="4987310">for</span> <span m="4987570">any</span> <span m="4988040">tournaments--</span> <span m="4988520">remember a</span> <span m="4988790">tournament from</span> <span m="4989260">graph</span> <span m="4989580">theory,</span> <span m="4989900">where everybody</span> <span m="4990520">beats,</span> <span m="4991410">you know,</span> <span m="4991820">between a pair of</span> <span m="4992690">players,</span> <span m="4993030">one</span> <span m="4993220">player</span> <span m="4993480">wins.</span> <span m="4994580">You</span> <span m="4994740">can</span> <span m="4994920">make</span> <span m="4996370">the</span> <span m="4996580">relation</span> <span m="4997200">for</span> <span m="4997420">any</span> <span m="4997680">tournament</span> <span m="4998130">show</span> <span m="4998380">up</span> <span m="4998810">for</span> <span m="4999000">a</span> <span m="4999050">particular</span> <span m="4999450">set</span> <span m="4999630">of</span> <span m="4999730">dice</span> <span m="5000110">by</span> <span m="5000230">rolling</span> <span m="5000630">a</span> <span m="5000690">certain</span> <span m="5000980">number</span> <span m="5001210">of</span> <span m="5001290">times.</span> <span m="5002770">All right,</span> <span m="5002970">which</span> <span m="5003220">is</span> <span m="5003310">just</span> <span m="5003550">really</span> <span m="5003870">fascinating</span> <span m="5004280">that you</span> <span m="5004690">can</span> <span m="5004780">prove</span> <span m="5005050">that.</span> <span m="5005280">So these</span> <span m="5005580">dice</span> <span m="5005910">can</span> <span m="5006030">get</span> <span m="5006210">arbitrarily</span> <span m="5007130">weird</span> <span m="5007660">in</span> <span m="5007760">terms</span> <span m="5008040">of</span> <span m="5008410">which</span> <span m="5008850">die</span> <span m="5009180">is</span> <span m="5009340">likely</span> <span m="5009640">to</span> <span m="5009700">come</span> <span m="5009910">up</span> <span m="5010440">and</span> <span m="5010550">be</span> <span m="5010900">better</span> <span m="5011540">than</span> <span m="5011670">another</span> <span m="5011920">die.</span> </p>
<p><span m="5013110">All</span> <span m="5013260">right,</span> <span m="5013520">that's</span> <span m="5013770">it</span> <span m="5013880">for</span> <span m="5013990">today.</span> <span m="5014280">Let</span> <span m="5014380">me</span> <span m="5014480">give</span> <span m="5014490">you</span> <span m="5014580">your</span> <span m="5014740">money</span> <span m="5014990">back</span> <span m="5015240">here.</span> <span m="5015430">And</span> <span m="5015550">I've</span> <span m="5015620">got</span> <span m="5015770">a</span> <span m="5015820">gift</span> <span m="5016020">certificate</span> <span m="5016610">you</span> <span m="5016710">can</span> <span m="5016810">have</span> <span m="5017365">for being</span> <span m="5017740">a</span> <span m="5017790">good</span> <span m="5017970">sport.</span> <span m="5020238">Here you go.</span> <span m="5020670">How</span> <span m="5021110">much</span> <span m="5021260">money</span> <span m="5021530">did I</span> <span m="5021620">take</span> <span m="5021890">here?</span> </p>
<p><span m="5022160">AUDIENCE: I think</span> <span m="5022300">only</span> <span m="5022770">like</span> <span m="5023150">$2.</span> </p>
<p><span m="5023740">PROFESSOR: $2?</span> <span m="5024280">All right,</span> <span m="5024550">there you go.</span> <span m="5025620">Very</span> <span m="5025920">good.</span> <span m="5026340">Thanks.</span> <span m="5029480">So</span> <span m="5029620">we'll</span> <span m="5029730">do</span> <span m="5029860">a</span> <span m="5029900">lot</span> <span m="5030130">more</span> <span m="5030280">examples</span> <span m="5030810">over</span> <span m="5030980">the</span> <span m="5031070">next</span> <span m="5031320">week</span> <span m="5031510">or</span> <span m="5031590">two.</span> </p>
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